How long do Oil Urns last?
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.