I think this is more of a movie thing.; I can't think of any drinking contest between a Dwarf and an Elf that takes place in Tolkien's legendarium. He does write in The Hobbit about the potent wines of the country of Dorwinion that are served at the Elvenking's feasts. Elves seem to be immune to natural diseases; perhaps they are highly resistant to poisons, making it hard to get them drunk.
Their livers can regenerate faster.