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I wrote this sentence (meaning "The temple is next to the park") in my Japanese course and my teacher said it's wrong, because I was supposed to use a は instead of a が.

My understanding is that there is always a が in a sentence, which may or may not be visible/dropped (zero-ga pronoun) and the subject marked by が may or may not be additionally marked with は. So I asked my teacher for clarification, what the subject of that sentence is and why using が was wrong, but the only explanation I got was "This sentence pattern uses a は", which doesn't really start explaining anything.

I tried to clarify, if using が here is merely unnatural (and that is what my teacher meant by "wrong") or actually grammatically incorrect, but my teacher assured me, that using が is ungrammatical in that sentence.

Can someone elaborate as to why が is indeed ungrammatical in this sentence? And if so, what is the subject of the sentence if not お寺?

In this case, が is incorrect because you are conveying a known piece of information. When you describe a known or general fact about a subject (お寺), you have to mark it with は, making it the topic of the sentence.

お寺は公園の隣です。
The temple is next to the park.
^{(This is a known fact to you.)}

鳥は飛べます。

Birds can fly.
^{(This is a general fact.)}

Note that the subject of the first sentence is still お寺, but now it's marked with は instead of が.

You have to use が when you firstly introduce something in the "universe of discourse":

お寺が公園の隣にあります。

There is a temple next to the park.
^{(As a response to something like "Is there a building related to Buddhism around here?")}

あの木の上に鳥がいます。

There is a bird on that tree.
^{(No one else has noticed this bird before this sentence.)}

You have to use が when you report a new event or temporary status you have just noticed regarding the thing marked with が. (neutral-description-が)

お寺が燃えています！
The temple is on fire!
^{(The listener knows which temple you are talking about, but the information ('on fire') is something you just noticed.)}

鳥が逃げました！
Our/The bird flew away!
^{(The listener knows which bird you are talking about, but the information is something new.)}

In addition, there is something called exhaustive-listing-が. With this, you are choosing and emphasizing something from several possibilities:

お寺が公園の隣です。

It's the temple that is next to the park.
^{(As a response to "So, what's next to the park is the post office, right?")}

鳥が飛べます。
Birds can fly.
^{(As a response to "Which vertebrate can fly?")}

This is probably the only case where お寺が公園の隣です would make sense, but if this is too difficult to you now, you can forget this and learn the easier ones first.

My understanding is that there is always a が in a sentence, which may or may not be visible/dropped (zero-ga pronoun) and the subject marked by が may or may not be additionally marked with は.

What do you mean by "the subject marked by が may or may not be additionally marked with は"? You can think は replaces が when the subject is topicalized, but you cannot mark a word with both が and は simultaneously. (は can replace を, too.)

using が here is merely unnatural (and that is what my teacher meant by "wrong") or actually grammatically incorrect

I would say it's grammatically incorrect (for your intended meaning). The distinction of は and が is critically important in most cases. People may guess what you are saying if you are lucky; in this specific case, people will probably understand you, but they never fail to think "Oh, this person is still bad at Japanese." If you are unlucky, it may end up with a severe confusion (example).

Related (if some of my explanation didn't make sense, please read these first):

• What's the difference between wa (は) and ga (が)?


• Can someone explain me the use of は and が in this sentence?


• Why does 「電話は切れた」 sound more adversarial than 「電話が切れた」?