It all depends on which superhero it is and how he's saving people. In general, I would say that the locality calls the superhero to help them or they've hired the superhero to be on call to save them, so they know that there may be some damage along the way.
Almost all the superheros are destroying objects that belong to the villains or these are objects that the villains use to try to harm the superhero. If someone throws a car at Superman, of course Superman can do anything he needs to save his own life.
Batman and Robin usually have their battles inside the villains studio or factory, so all the property belongs to the villain. It's rarely public property. Sam concept, here - the villain is using the chain to lower the Dynamic Duo into the hot acid. Of course they can destroy the chain, the vat, etc. anything to save their own lives.
As for action shots like in Robocop - all the cars and motorcycles are bought by the movie studio and they get advertising perks from car manufacturers too. They specifically tell them to bounce the car / cycles; make them fly in the air; blow them up; do whatever you want. Nothing will happen to these things - they'll still drive well all dented and with a burned engine. That's the whole point to it! They hope that people like you and me watching such strong vehicles being beaten up will end up buying these cars that can fly in the air like that! So, no, they don't have to pay for any damages to them.
As for any other occurrences, the evidence doesn't appear to warrant a necessity for paying damages. It looks like the town is all fixed up fine in the next scene, and the superhero is on to saving the next person, or city. They've probably been reimbursed by either the movie studio or the superhero's agent.