This came up in a session recently, and we couldn't really decide on how this would work. Basically, I wanted to move through an ally's square, and then shove them backward... so I could move up to the front line. Is this possible?

Shoving requires a contest, but if the creature doesn't contest, then would it be an auto-success?

Yes, you can shove an ally

Shoving

The rules for shoving say

Using the Attack action, you can make a special melee attack to shove a creature ... The target must be no more than one size larger than you and must be within your reach. (PHB, p. 195)

There is nothing here to suggest that the target must explicitly be an enemy. Your ally is a creature, so you can attempt to shove them (provided they are no more than one size larger than you and within your reach).

Even if the rules did imply that the target should be an enemy, the rules never define the distinction between friend and foe. You could as a player declare your 'ally' to be an enemy after you have moved through them but before your shove, then declare them to be your friend again afterwards (assuming they accept your offer of friendship). But this should be unnecessary unless your GM is oddly adamant that you can only shove an enemy, and if they are of that persuasion they would be unlikely to let you switch your ally-enemy designations so easily.

A tactical reason to shove your ally

There is an advantage to moving your allies using the shove action (provided you can get the correct positioning to shove them in the correct direction): your ally won't be affected by opportunity attacks.

You also don't provoke an opportunity attack when ... someone or something moves you without using your movement, action or reaction. (PHB, p. 195)

Getting shoved does not use your ally's movement, action or reaction, so your ally does not provoke an opportunity attack when shoved out of an enemy's reach.

Movement

From 'Moving Around Other Creatures' (PHB, p. 191), you can move through a nonhostile creature's space, although it counts as difficult terrain so costs double movement. This means you are free to move past your allies. However,

Whether a creature is a friend or an enemy, you can't willingly end your move in its space.

This means you cannot stop in your ally's space to shove them. There must be an empty space on the opposite side of your ally for you to move into before you can shove them.

Contest

As for whether your friend can willingly and automatically fail the shoving contest, the rules-as-written say no, since no provision is made for willing failure in the rules for shoving or contests. However, the rules-as-written would allow the DM to provide advantage to you and disadvantage to your friend if you are both working in unison to shove your friend. Additionally, as the defender of a shove can choose between Strength (Athletics) and Dexterity (Acrobatics), your friend could choose the worse of those two skills to further increase the odds.

There are some other alternatives. From this question on failing ability checks, the general consensus appears to be that while there is no provision for choosing to automatically fail a roll, you could attempt a different check where success would be your desired outcome - however, this is no longer the official Shoving a Creature action, but some improvised action which your DM would have to approve and determine DCs for (if success if not decreed to be automatic). Additionally, by not using the Shove a Creature action, you probably lose the benefit of being able to substitute a single attack for a shove and instead need to use your whole action.

Rather than trying to shoehorn success out of a strict rules-as-written reading of the shoving rules, your DM could employ common sense and simply declare that you succeed automatically. As pointed out by KorvinStarmast, the text under 'Ability Checks' (PHB Chapter 7) says,

The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.

Shoving uses ability checks so can be argued to fall under this rule, although to make things ambiguous shoving also uses your Attack action (but not an attack roll). It would be within reason for the DM to rule that shoving a creature who is not resisting succeeds automatically, regardless of the minutiae of the rules.

You can try

The play of the Dungeons & Dragons game unfolds according to this basic pattern:

1. The DM describes the environment.


2. The players describe what they want to do.


3. The DM narrates the results of the adventurers’ actions.

If you want to step forward and push an ally back, as a player you describe what you want to do — for instance, "I step forward and push Olaf back as far as I can". It's the DM's job to adjudicate this action and resolve the outcome, maybe with asking for some ability checks in the process. The checks being asked, as well as the possibility itself might depend on the situation.

We can not guarantee you success. D&D 5th edition empowers the DM in ways that 3rd, 3.5, and 4th did not. While rule zero has always applied, 5th edition chooses not to explicitly codify many things. If your DM says you can't, you can't.

Anything you can do to an enemy, you can to to an ally

Well, almost. There are very few features (like Paladin's Holy Nimbus) that actually affect enemies only. As a general rule tho, any action that affect enemy can affect ally as well. You can attack an ally. AoE spells harm allies and enemies likewise, regardless of the caster's intention. But not vice versa — there are things allowed for allies only.

Keep in mind, that "ally" and "enemy" are not strictly defined game terms. The playbooks use the term "willing creature" instead of "ally", which is a little more specific. Aside from being willing, there are little or no mechanical differences between allies and enemies in 5e.

Possibility to intentionally fail a check is table dependent

Although it is more or less clear, there are no strict rules for failing checks on purpose. DMs are suggested to use the common sense here — if a check represents something you choose to do intentionally, you can naturally choose not to do that. A good DMs' practice is asking about the opposing side intentions ("Olaf, do you want to be shoved?") before choosing the respective mechanics. However, a DM can house-rule that and allow (or forbid) intentionally failing for all checks and saving throws required by the rules, for the sake of simplicity.

Depends on positioning

If you are trying to replace a character on the front lines then the problem is how the shoving rules work. These rules state:

Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.

The target must be no more than one size larger than you and must be within your reach. Instead of making an attack roll, you make a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you either knock the target prone or push it 5 feet away from you.

The text is quite clear in that you are pushing the creature away from you, not repositioning them. If your ally is in front of you and there an an enemy in front of them, then you'd be pushing them into the enemy. You can't pull them back out of the front line and take their place. You also can't move into their space and shove them behind you due to the rules on moving around other creatures, which state:

Whether a creature is a friend or an enemy, you can’t willingly end your move in its space.

Your DM may allow you to do this anyway though as they are the final arbitrator of the rules and it would make sense to allow it.

On the other hand, if you are trying to simply move your ally to a different position then you can use an attack as a shove. However the same rules apply, you are shoving the other character directly away from you, not moving them to a specific location.