The fifth edition of Dungeons & Dragons has no official rules for getting drunk, but fans have created house rules for their own games. Expect to take some ability roll penalties, especially if you have a low Constitution.
If your party is visiting an especially well-appointed tavern, such as Little Chicago Entertainment, then you may have options beyond the conventional ale and wine. This establishment in Moose Jaw, Saskatchewan offers boardgames, axe throwing, and alcohol. If you’re not sure what to drink, roll a 20-sided die. Redditor u/AnaphlacticHippo shares this photo of a random drink menu. It could be improved by making a critical failure (a natural 1) result in a double.