“会”在这里表示“能够”“可以”，是一种能力。

我不喝酒Usually it means I can drink alcohol but I don't drink it now. 我不会喝酒 I can't drink alcohol.

Well, you have good reasons to doubt the use of 会， as it does sound weird to a native ear linguistically.

Wow, this is too much Chinese for me :D I literally just started recently and this sentence came up. Nevertheless, I appreciate all of your input and I think I understand how it's used now :)

Hi Rebecca, so you find it weird to say 我不会喝酒，as 喝酒 is such a no-brainer: just pour the liquid into your mouth and gulp it. How can someone be technically unable to do that? Don't worry, I can relate, I feel exactly the same with you, although I am a native speaker of Mandarin. Every time they say this, my linguistic ego feels annoyed. But I settled. I see it as euphemism: it would be, in some contexts, too abrupt to say 我不喜欢喝酒 or even 我就是不想喝，你管得着吗？ For deeper understanding of this issue, you may want to know something about 白酒， and 劝酒文化。Note that 白酒 is much stronger than beer, and not every Chinese is a big fan of it.