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Looked and looked, can't find the answer anywhere (maybe I'm blind):
How long does Alchemist's Fire burn for?
In the item description within the Adventuring Gear section of the rules, it says:
On a hit, the target takes 1d4 fire damage at the start of each of its turns. A creature can end this damage by using its action to make a DC 10 Dexterity check to extinguish the flames.
Yet no mention of the burn rate. One round after a hit? 2? A minute? Continuous until extinguished?
Alchemist’s Fire.
... On a hit, the target takes 1d4 fire damage at the start of each of its turns. A creature can end this damage by using its action to make a DC 10 Dexterity check to extinguish the flames. (Basic Rules, p. 47; PHB, p. 151)
If no effort is made to extinguish the fire, it will keep burning. The descriptive text does not have a time limit. Compare that to oil (flask) which does:
If lit, the oil burns for 2 rounds and deals 5 fire damage to any creature that enters the area or ends its turn in the area. Basic Rules, p. 52)
The attack can miss, however, so it's not a guaranteed problem for the target.
Make a ranged attack against a creature or object, treating the alchemist’s fire as an improvised weapon
The description of Alchemist's Fire covers only one use case: throwing the flask at a creature that you want to set on fire. The only described way for the fire to go out is for the creature to extinguish it. (Note that extinguishing it doesn't require magic or special tools, just dexterity, implying that it is a mundane fire that needs air.)
Other than that, the rules say nothing about how long the fire keeps burning.
