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.)
Alchemist‘s Fire Edit Page Content. This sticky, adhesive fluid ignites when exposed to air. As an Action, you can throw this flask up to 20 feet, shattering it on impact. Make a ranged Attack against a creature or object, treating the alchemist‘s fire as an Improvised Weapon. On a hit, the target takes 1d4 fire damage at the start of each of …
Alchemist’s Fire – Lord of the Craft
Ignited Alchemist’s Fire can get hot enough to melt silver and soften stronger metals such as steel, and is capable of sustaining itself in scenarios that natural would be incapable of, such as directly on or being covered in water, as the air in the water is still enough to keep the fire going and not be completely snuffed out.
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Reference:
https://www.usfa.fema.gov/
https://en.wikipedia.org/wiki/Fire_extinguisher