Is it possible to make a beverage that evaporates on contact with the consumer's tongue?
In the Star Trek world you have plenty of beings with body temperatures higher or lower than humans, so if you had a being with a body temp of 79C then ethanol would work out perfectly! However, with humans the liquid which would fit the bill of vaporizing just below body temperature would be Pentane, which not only smells bad but is also toxic. The science behind it is wrong anyway, you'd want a liquid with a much lower vaporization point than body temperature so it would boil off quickly. If you pick a liquid that boils just below body temperature it would take some time to vaporize and you wouldn't have the same effect.
All is not completely lost, you can still have some cool effects with liquids that are safe for consumption. What you have to remember is that the closer a liquid gets to its boiling point the more it evaporates, and any volatiles in it will tend to remain suspended in the vapor and be detectable by the nose. This is the effect you get when you sniff brandy or cognac that has been brought up to body temperature. You can savor it by holding it on the tongue for a few seconds and letting the vapors fill your nasal cavity. So all you need is some strong (at least 80 proof) and flavored (or more accurately scented) alcohol, and the right glasses. Brandy snifters would work best as they are designed to capture the vapors so they can be inhaled. You could then heat the liquid to just above human body temperature and then serve.
Keep the quantities served down unless you want some very drunk guests!