It was noted below that this is properly called the partitive case; the Wikipedia articles have good examples in the vein of "А не испить ли нам чаю":[1][2]

[1]https://en.wikipedia.org/wiki/Partitive_case#Russian

[2]https://ru.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D1%82%D0%B8...

The partitive here is to express that the speaker is not specifying in what form or shape the beer in the fridge should appear. For a native speaker it sounds like you are talking about some quantity of beer but omitted the quantity so it's literally just some unspecified quantity of beer.

So you would say that though? Like I've just never heard my хозяйка квартиры say that is all. I hear it often in things like "would you like some tea/beer?" etc or "i'm going to buy some tea/beer" but never like that. My impression was that outside of those contexts it's not popular. Mind sharing some examples? Would be really useful! :)

You probably would never hear somebody saying "Поставь пива в холодильник." since, while a correct sentence, it would need some rare circumstances to be said. Поставь here is much more concrete than "put", it's more like "place" and I could imagine saying this only if I had been managing the fridge at some party and noticed it's running out of beer so I requested somebody to add more beer there. Much more common would be "Принеси/достань/возьми пива из холодильника" (get some beer from the fridge).

>You probably would never hear somebody saying "Поставь пива в холодильник." since, while a correct sentence, it would need some rare circumstances to be said. [...] I could imagine saying this only if I had been managing the fridge at some party and noticed it's running out of beer

Yup, pretty much that. Or like that one time when we went binge-shopping on craft beers, but there was only enough space in the fridge for a couple of bottles; so that was a normal request about an hour before we were to watch a movie. Which is pretty much the exact situation you described :)