Most debris from a remodel, roof, or cleanout can go in your container. California's trash rules and the landfill set the limits.
CalRecycle keeps a list of wastes banned from the trash, and none of them belong in a dumpster. The list covers batteries, fluorescent tubes, latex and oil-based paint, solvents, antifreeze, motor oil, asbestos, and treated wood. Tires are on it too, to prevent mosquito breeding and tire fires.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster. Electronics also need their own route, so keep TVs and computers out.
The City hosts household hazardous waste events on the first Saturday of most months, which is the right place for paint and chemicals. If you're unsure about an item, check with your account manager first.