Here's the description of an oil flask:
Oil usually comes in a clay flask that holds 1 pint. As an action, you can splash the oil in this flask onto a creature within 5 feet of you or throw it up to 20 feet, shattering it on impact. Make a ranged attack against a target creature or object, treating the oil as an improvised weapon. On a hit, the target is covered in oil. If the target takes any fire damage before the oil dries (after 1 minute), the target takes an additional 5 fire damage from the burning oil. You can also pour a flask of oil on the ground to cover a 5-foot-square area, provided that the surface is level. 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. A creature can take this damage only once per turn.
The description provides a duration that oil burns while poured onto a surface, but not for oil that's lit while splashed onto a creature. Does a creature covered in oil take the additional 5 fire damage every time it takes fire damage for the one-minute duration until the oil dries, or only the first time?
5e is usually pretty literal about this, and it doesn't say 'the first time the target takes fire damage,' so I'm inclined to think it's the latter. But that seems pretty powerful for adventuring gear that costs a single silver piece, even if it does take an action to set up.