No, you can't light it on fire without ending the conjuration, and still be within the rules as written. The spell doesn't specify that it ends "if [the object] takes 1 hit point of damage", it specifies that it ends "if [the object] takes any damage." Any damage definitely includes burning, tearing, scratching, cutting (with teeth), etc. — and depending on your DM,1 may also include splitting the object into multiple drops or any other alteration that either is irreversible or would split the object into multiple objects.
No, it's not overpowered played otherwise. It's just not something that Minor Conjuration allows for without deliberate house rules.
I wanted to address another issue in the question that you didn't ask about, but is also contrary to the ability's scope: You can't summon anything above a creature's head. It must be "in your hand or on the ground", and further, if conjured on the ground it must be in an unoccupied space. There's no way to get this puddle of oil within danger-range of a creature, even if you could light it.
- The ability specifically deals with "an object", not "a volume of substance" as some other abilities in the game do. That means that a DM is within their rights to rule that a puddle of X is the object, not the volume of substance the puddle is made of, so damaging the puddle will end the conjuration. This is probably something most DMs won't make a big deal about, but a DM would be following literal RAW if they had a conjured puddle disappear if any of the puddle was removed. Different DMs will rule this differently, because the real ontological nature of anthropocentric concepts such as "damage" and "object" — is a page, cut up, damaged? is a glass of wine, drunk, damage? is a glass of wine, dropped on hard stone floors, damaged? if the answers are different, why? — is an unsolved problem in metaphysics.