Here's my perspective.
Dwarves drink beer, obviously. They spread the joys of beer every year with Brewfest.
Night Elves go right for the Franzia. It makes them feel classy.
Blood Elves, on the other hand, drink colorful, fruity things and chase them with shots of mana potion.
Goblins? Four Loko. And then they blow stuff up.
Trolls go for stuff like rum occasionally, but prefer stuff you can smoke rather than drink.
The Forsaken just go for whatever burns the most - and it's a not-so-quiet secret Sylvanas is also nicknamed "Vodka Queen". The statue in Brill? Ordered while drunk.
Humans will drink damn near anything, but they do like their rum in Theramore.
Worgen like to think that drinking champagne makes them classy, but then they wake up in the morning surrounded by empty bottles and wearing nothing but a top hat.
Draenei go for Hypnotiq. It comes in blue and purple, that’s why.
Gnomes probably drink Four Loko, too, now that I think about it. Or schnapps.
Orcs will also drink anything, and they think humans are wusses for doing shots of tequila instead of chugging it.
Tauren like cider. And White Russians, but they get enough crap just for drinking plain milk, so they stick to cider.
Pandaren like beer, too, but once they get off the Wandering Isle and join the rest of Azeroth, their new buddies are going to make them try EVERYTHING and they are going to get SO SMASHED, it will be hilarious.
Murlocs, of course, drink martinis. Shaken, not stirred.