My daughter ran out of oil in her new apartment. This has happened just once. She got oil and then the furnace ran fine for a couple of days. Then she had to start using the reset button. Landlord had a tech come look at it. Turns out the furnace needed a new pilot light. Landlord says it was because they ran out of oil. Can running out of heating oil one time cause pilot light to need replacement? The landlord says it is her fault.
Thanks... I don't know much about furnaces!

Anything is possible. The question you are asking has no real answer. If the unit needs to be fixed it has to be done. I will say that running out of oil usually just means they restart the machine but it probably needed a tune up anyway.

Maybe I haven't seen it all yet? An oil furnace with a pilot light? Can somebody else help me?

Oil furnace would have a burner moter, nozles, flilter and an electric ingnitor. Running the unit out of oil can cause "trash" in the botom of tank ( depending on how tank is plumbed) to get into filter. Also it can cause problems with the oil pump. And it can most certainly burn the ignigtor out

So yes, running a oil burner out of oil can cause serious and expensive repairs.

Who's responcibility is it to maintain the furnace? Yes running out of oil will cause a service call but not necessarly a complete cleaning which should be done at least once per year.