Background

I have a 9th Level Oath of Redemption Scourge Aasimar Paladin. Due to the way the story has evolved, I will need to multiclass into warlock. My intention was to take levels 10, 11 and 12 into 3 Levels of Warlock and pick up the Celestial Patron with Pact of the Blade.

This character is the only divine member of the party. As such, he is the primary tank and primary support.

Question

I know that I'll be losing the Paladin's Aura buff, but beyond that I see most people suggest Paladin 18 / Warlock 2 or Paladin 16 / Warlock 4. Why is this? What am I losing by taking 3 levels of Warlock instead of 2 or 4?

Primarily I'm valuing the Paladin's 5th level spell list as the reason to only take exactly 3 levels in Warlock.

TL;DR: A 3 levels dip is probably your best option, as it more than compensates for the abilities you will lose as a paladin considering your role in the party. Warlock 3 grants you two extra 2nd level spell slots (renewable on a short rest) that can be used to smite or heal yourself, without costing you the ability to cast 5th level spells. Celestial pact more than compensates for your lost Lay on Hands HP, and Pact of the Chain is likely your best option as it grants you, through the Gift of the Ever-Living Ones invocation, the ability to maximize every die you roll to heal yourself, making you a most formidable tank.

People put a lot of value on warlock 2 as it gives you access to invocations. If your objective is to become an eldritch blaster, that's basically all you need, i.e. access to eldritch blast and agonizing blast. Similarly, warlock 4 means that you don't lose an ASI, which can be highly valued in cases where multiclassing puts you in a tough spot regarding ability scores. Both don't apply so much to you, as you are most likely not going to spam eldritch blast given your role inside the party, as well as the fact that paladins usually possess high charisma scores. Warlock 3, on the other hand, isn't generally considered to be very useful, as the pact boon is usually wasted when multiclassing. Those are mostly good for lower level characters unless you "scale" them through invocations at later levels.

However, in paladin's case, especially one that enters the warlock class at much later levels, I think warlock 3 is most likely optimal for many reasons. You also mention being both the tank and support for your party, so we'll compare what you gain/lose by multiclassing by taking a closer look at the following subsets of your abilities : healing, spellcasting, offense and defense, as well as analyzing your different options regarding ASIs, Pact boons and invocations.

Healing abilities

The Celestial patron's healing light feature does a good job of compensating for the lost lay on hands HP. As far as your healing abilities go, you do not really lose anything by multiclassing into warlock, except that you can't use all your healing in one shot if the need arises (lay on hands and healing light need to be used separately). This feature can also be used up to 60 feet away, meaning that, unlike Lay on Hands, you can prevent an ally from dying without having to relocate, potentially avoiding opportunity attacks in the process.

Since the amount healed from healing light is 1d6 + 1d6 per warlock level, this gives you a slight healing boost at 1st level (7HP avg) and evens out at 2nd level (10.5HP avg). A 3 levels dip (14HP avg vs 15) and a 4 levels dip (17.5HP avg vs 20) represent slight nerfs to your healing abilities on average, but not in a significant manner.

If healing is your primary concern, it doesn't really matter how big your warlock dip actually is, though you get slightly more potential healing with a 2+ levels dip, because of the second warlock spell slot that could be used for an extra cure wounds spell. Some invocations (as we'll see later), can also turn this into a huge buff to your healing abilities.

Spellcasting abilities

A four level dip costs you the ability to cast 5th level spells, which is a heavy price to pay considering you only gain 1 cantrip (of which you already know 4) and one 2nd level spell known. Considering you are the only divine member of the party, the inability to cast 5th level spells means that no one in your party can cast raise dead, for instance.

If you wish to maximize spellcasting abilities, I recommend you dip 3 levels into warlock. This allows you to keep access to 5th level paladin spells, as well as giving you 2 extra renewable spell slots of 2nd level from warlock. The spells you get from warlock can also be very support-oriented. Spells such as Invisibility and Spider Climb can give a great edge to your rogue when sneaking around, for example. Comprehend language can give you an edge in social challenges. Hold Person can be great crowd control. Etc.

Basically, multiclassing into warlock gives you more spells known, and many of those wouldn't normally be available to paladins. As far as spellcasting goes, a 3 levels dip is a buff to your spellcasting versatility and potential, though it delays your progression a little bit.

Offensive abilities

Besides the fact that you delay the moment at which you gain improved divine smite, which is a very good damage increase for paladins, you do not really lose anything in the long term. Given your status as the main tank rather than a damage dealer, this delay should be acceptable.

In fact, multiclassing into warlock might actually increase your damage output in situations where you can't immediately enter melee range, as sacred flame and eldritch blast both scale very decently. This means that you don't need to lose actions to stow your sword and shield away and draw a ranged weapon: you can just use those cantrips while closing range. Also, if you take the agonizing blast invocation, eldritch blast's damage could actually compete with your regular melee damage.

Defensive abilities

The loss of your aura's extended range isn't as steep a cost as it sounds. In general, ranged characters will likely be farther away than 30 feet from you anyway, meaning that the extended range only facilitates positioning for other melee characters in your party. Nonetheless, it's an ability you gain extremely late, and your party's melee characters should already be used to the required positioning for your auras. So while it's a very nice buff to have, losing it doesn't suddenly break the paladin class as your party can still fully benefit from your auras just as they've been doing for over 12 levels.

Outside of that, the only defensive drawback is, as stated above, the inability to use 100% of your healing powers in one single action if you need to. Otherwise, you still heal a lot of HP, some of them are just a bit more random.

On the other hand, getting access to the Armor of Agathys through multiclassing tremendously increases your defensive powers by giving you extra hit points and damaging enemies that strike you. Since you can cast Armor of Agathys using your paladin spell slots, this can mean an extra 20 or 25 hit points for the fight, as well as potentially 50+ damage dealt to enemies. Basically, this spell alone is a better use for your spell slots than smiting, as it does more damage as well as providing great defense. Armor of Agathys requires no concentration and lasts for one hour, meaning that you can even combine it with mirror image, also available to warlocks, for further defense. Or perhaps with shield of faith.

At any rate, you will lose the Emissary of Redemption class feature, but proper application of Armor of Agathys will give you very similar benefits for a time (temporary HP instead of resistance and returning damage to your attackers), all without losing the benefits if you actually attack them during the fight (unlike Emissary of Redemption).

Ability score increases

If your campaign actually goes to level 20, the loss of a single ASI shouldn't matter that much. At this point, you should already have the feats you want and 20 in your primary stat, and likely 18 or 20 in your secondary stat. Like the aura buff, it's nice to have, but it shouldn't break your build if you don't get it.

Pact boon

This one here is somewhat tricky. Basically, as you're not taking more that 4 levels in the warlock class, none of these abilities scale in any significant fashion. You mention going for pact of the blade, but this would merely grant you the ability to store your (likely already magical) weapon in an extra dimensional space. Pact of blade is generally useful for single class warlocks, as it's the only way to get an extra attack (5th level invocation) and extra charisma-based damage (12th level invocations). You already received similar features as a paladin (at 5th and 11th levels respectively), so pact of the blade is likely not going to give you anything special. Similarly, given that your patron already grants you 2 extra cantrips, the pact of tome doesn't really do much for you unless you really want a specific cantrip from a different class.

I would personally suggest going with pact of the chains, which would allow you to get a cool familiar such as a pseudodragon or a sprite (assuming you don't want the more evil-themed familiars). Most notably, the pseudodragon has blindsense, allowing him to easily detect invisible creatures within his limited reach, as well as advantage on pretty much every perception checks (sight, hearing and smell). If your DM allows it, you could also potentially harvest his poison. The sprite is also interesting, as it can fly and turn invisible at will, allowing it to scout for you.

More importantly, however, Pact of the Chain is mostly valuable for its access to invocations...

Invocations

Since you can't smite on your cantrips, maximizing you eldritch blast damage is not necessarily what you want to do. You only get two invocations, so you should make them count as much as possible.

As you have access to Xanathar's Guide to Everything, the Gift of the Ever-Living Ones stands out as a very interesting option. It requires the Pact of the Chain feature, but allows you to maximize any die rolled to heal yourself as long as your familiar is within 100 feet of you. Healing light? Maximized. Cure Wounds? Also maximized. Taking a short rest? YES! So yeah, that invisible sprite that no enemy actually knows is there is like an actual mini guardian angel sent by your god to make you invincible.

If you're taking blade pact in the end, the Improved Pact Weapon invocation allows you to use your weapon as a focus for your warlock spells. However, you're most likely to use these spell slots to either smite or heal yourself using cure wounds (which you already know as a paladin), meaning that your holy shield already covers most of that for you.

As far as supporting abilities go, certain invocations can grant you extended darkvision that pierces even magical darkness, the ability to detect magic at will, to read any language, to cast disguise self at will, to talk through your familiar, etc. Depending on what you want to do, there are many great options, though if you decide on going for pact of the chain instead of pact of the blade, Voice of the Chain Master has interesting strategical applications, as it allows you to send your invisible fairy scout with the rogue and discuss your plans as if they were still with you.

Conclusion

If you're dipping in warlock, you should always take at least 2 levels for the extra spell slot and access to invocations. Losing an ASI isn't fun, but you stand to gain much more power from the dip that it shouldn't really matter. Celestial warlock combined with Pact of the Chain makes you into a formidable tank, allowing you to maximize every die you roll on self-healing, making a 3 levels dip very appealing. Dipping 4 levels into warlock doesn't grant you much as far as warlock abilities go, though it gets you your ASI back, but it does cost you 5th level paladin spells, meaning no Raise Dead of similarly powerful spells.

All things considered, I think a 3 levels dip into warlock is your best option. The only thing you really lose is the extended auras and delayed access to some abilities. On the other hand, the proper invocations/pact boon will greatly enhance your abilities as a healer/tank, more than compensating for this loss.

You lose the paladin features of levels 17,18,19, and 20 depending on the dip

Functionally, this is pretty big. You not only lose some big paladin abilities, but you also delay your standard progression. I'm going to first go through what you things will be missed and will then cover my experience playing a full paladin as well as my experience in doing a multiclass warlock dip from Bard.

You will specifically lose:

Two level dip

1. Your capstone


2. An ASI opportunity


3. A 2nd 5th level spell slot


4. Loss of 10 Lay on Hands HP

Three level dip

1. Your capstone


2. An ASI opportunity


3. A 2nd 5th level spell slot


4. 
   Increasing your Aura abilities to 30'


5. Loss of 15 Lay on Hands HP

Four level dip

1. Your capstone


2. ASI delayed


3. 
   Both 5th level spell slots (No access to any 5th level spells)


4. 
   Increasing your Aura abilities to 30'


5. Loss of 20 Lay on Hands HP

Considerations

You ask what you will lose and that's been stated above. Whether or not losing those is worth what you get from the dip is entirely up to you - but you definitely lose some big things.

You'll also have to assess what things your party can bring in both their racial abilities, class abilities, and through magic items provided to you by the DM.

You'll also need to consider how many combat encounters you have per day and how that fits into your build(s) as to which may be best for you as a player, your character's in-game progression, and the party makeup.

I've played a paladin from 5-20 and also done a warlock dip as a bard. I'm going to use those two experiences to talk about dipping a bit.

Straight Paladin

The Paladin capstones are generally a very big thing. And the Redemption Paladin's is included:

At 20th level, you become an avatar of peace, which gives you two benefits:

• You have resistance to all damage dealt by other creatures (their attacks, spells, and other effects).


• Whenever a creature hits you with an attack, it takes radiant damage equal to half the damage you take from the attack.

If you attack a creature, cast a spell on it, or deal damage to it by any means but this feature, neither benefit works against that creature until you finish a long rest.

Keeping yourself alive is paramount, especially if you're doubling to keep people alive. Your capstone not only basically doubles your HP by giving you resistance to all damage, but that damage that does get delivered is returned by half to whomever dealt it. Even if and when you get it, you're still causing damage and potentially taking someone out of that. Sacrificing that alone is a big loss.

The loss of Lay On Hands is also a big deal. It varies depending on the dip, but when all you need to do is spend an action and touch someone to bring them back, the more HP you have to do that, the better.

The ASI delay may or may not be a big deal depending on your build, but losing it might hurt a fair amount more.

Allowing your Auras to extend to 30' also further protects your allies. The Guardian aura is another opportunity to mitigate damage and may help in keeping your allies up (and you can utilize your Lay on Hands to help ameliorate the damage you've just taken.)

And finally, while you can't use 5th level slots for more smite damage dice, you can still use them as smites or as other spells. Holy Weapon was one of my favorites, but there are other supporting auras or other things you can do with those slots that can make a huge difference in keeping your teammates alive.

Bard with Warlock Dip

I had done this for roleplay purposes, but also to give myself another attack option when I was concentrating. I loved having htat option, but delaying my Bard progression was a big deal. And depending on when you take your warlock dip for Paladin, you'll be delaying at some point some of the big paladin class options (including bigger smites and other paladin class abilities.)

Dipping also delays your primary class progression

I personally found that I felt frustrated how far behind I was in my bard progression and how it hurt me. The warlock things I got were nice, but by the end, I wished I hadn't done it.

What about you?

Well, only you can decide if the trade-offs will be worth it. I've provided you with the list of things you'll lose that you requested and given a couple of anecdotes about my own experience in multiclassing with warlock and in not multiclassing paladin.

Either way, enjoy the ride with whatever decision you make. The good news is that it all can be fun :)

It depends on how long the campaign runs

You've asked about Paladin18/Warlock2 vs Paladin17/Warlock3 vs Paladin16/Warlock4, but that seems like a weird thing to focus on.

Many games don't even reach level 20. (The game gets harder to run at high levels, so games that are stable initially might not remain stable.)

Of the games that do reach level 20, most of them reach a final episode and end shortly after that. There are basically no games that reach level 20 and then keep playing at level 20 for a long time.

So the real question you should be thinking about is not "what will be the best character for the 5% of the game when we're exactly level 20?" but rather "what will be the best character for now?"

(Source: I've played in a lot of games, and in some of them I tried to make decisions that would give good results at very high levels, and none of them got to very high levels. I tried optimizing for the current level instead and I was happier with that.)

With that in mind, the question becomes easier to answer: you should look at those warlock levels you're gaining, and think about whether each of them is worth delaying your access to your next few paladin levels. At warlock level 4 you're getting an ASI that you probably don't super need; at warlock level 3 you're getting Pact of the Blade. Pact of the Blade is a great way to get a magic weapon if you need one, but it seems like by tenth level you should have one of these already.

In exchange, you're slowing down your access to paladin level 11 (+1d8 radiant damage per attack). You're slowing down access to other stuff too, of course, but that's the most immediately relevant thing.

I'd speculate that the Paladin16/Warlock4 build is intended for people who dip Warlock very early and want to get the warlock ASI. The Paladin18/Warlock2 build is intended for people who are optimizing for combat and don't need the familiar or the book-of-cantrips from Warlock level 3. Your Paladin17/Warlock3 build seems like it's focused on giving a lot of versatility but combat power isn't particularly a priority.

Good luck with it.

I'm going to suggest a substantially different level split:

You should take 8 levels of Warlock

You say you value 5th level Paladin spells highly, but you should not. They are weak.

Think of them this way: A 10th level Bard (with Magical Secrects) can cast any spells you could on 17th level. It is not owerpowered for him, so it must be weak for you.

Paladin 12 / Warlock 8 is better

+5d8 twice per short rest is stronger in my opinion than anything you could get from Paladin after level 12.

You got Improved Divine Smite and ASI, great place to leave.

What you lose

Protective Spirit is good, but fights usually do not last that long. At the end of the first turn you are likely over half HP, and one or 2 rounds later the fight is already over.

Cleansing Touch can be really useful under the right circumstances.

Aura of the Guardian with 30 feet radius is great, but only marginally better than with 10.

Party members who perfer to be in melee can remain close to you, and the others should be more than 30 apart if possible, as it is the walking distance of most monsters.

What you gain

The two 4th level slot per short rest is already mentioned.

Cantrip: One more cantrip over Warlock 3

Spells: Counterspell¹, Dispel Magic, Fear, Fly are all quite good on a frontliner.

Invocations: Another two over what you could have with 16/4.

Radiant Soul: + Cha damage to Branding Smite, and some other, less useful spells.

1) You have to be within 60 feet of the enemy casting the spell, so you are generally in better position than your caster colleagues