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 :)