Most remodeling and cleanout debris can go in your container. California's disposal rules and the facility that sorts your load set the limits.
CalRecycle's list of waste banned from the trash includes fluorescent tubes, LED lights, and every kind of battery. It also bans latex and oil-based paint, thinners, aerosol cans that aren't empty, pesticides, antifreeze, motor oil, and filters.
Tires are on the list too. They aren't hazardous, but California keeps them out of the trash to prevent mosquito breeding and tire fires. Older homes need extra care, since asbestos can show up in old cement, roofing, flooring, and siding. Treated wood and propane tanks are banned as well.
Electronics, fuel, and liquids don't belong in a dumpster either. If you find any of these while you clean out, set them aside for a household hazardous waste drop-off.
When you're unsure about an item, your account manager can check before it goes in.