How long does an Oil Urn last after hitting an enemy with one? And also if I used a Molotov on them after covering them in oil, does the oil burned away after inflicting more damage or does it stay until they're dead?
I can confirm that Oil is good only for the next Molotov.
I tested out the question on a boss. An ordinary Molotov did around 135 damage. An Oil Urn directly on-target, followed by a Molotov, did about 285 damage. The next Molotov went back down to 135.
It appears it's akin to Bone Marrow Ash for 1 QS bullet shot — I.e. it's good for 1 Molotov.