How would you handle this? I'm thinking it will do max damage, including splash damage, so 14hp per round for two rounds. Additionally, I think it will prevent that player from casting spells with the Vocal component until fully healed. Too much? Too little?
It's a matter of what you mean by drinking it.
If the character is just uncorking the bottle and taking a swig as though it were wine or something, it will probably ignite soon after the cork comes out, or possibly as it goes into his mouth. He'll probably then attempt to spit it out (DC15 Reflex?). If successful, I'd say 1d6 damage, fire in mouth goes out, bottle is still burning, no further effect. Otherwise 2d6, try again next round. Apply the total damage to a concentration check when casting Verbal spells until healed.
If he's really trying to drink it all down, knowing what it is, he should attempt to stop it igniting by excluding the air. DC20 to do that. If failed, see above. If successful, it's probably really nasty (DC15 Fortitude to get it down) and somewhat toxic (DC15 Fortitude or be nauseated for 2d6 minutes, sickened for 2d6x10 minutes).
In this case, it is a Princess Bride-like challenge. The unobservant character may end up drinking an alchemist fire as it is oddly bottled, including swallowing it.