Most remodeling and cleanout debris can go in your container. California's disposal rules and the City's recycling rules set the limits.
California makes it illegal to throw hazardous waste in the garbage. According to CalRecycle, the state's recycling agency, that covers all batteries, fluorescent tubes, LED lights, motor oil, antifreeze, oil-based paint, and aerosol cans that aren't empty. Electronics are on the list too, including TVs, monitors, laptops, printers, microwaves, and cell phones.
A few building materials also stay out. Treated wood has to be handled as hazardous waste in California. Tires are banned from the trash, and so are propane tanks and other gas cylinders. Asbestos and older materials with PCBs, like some caulk and light ballasts, need special handling.
Liquids, solvents, and fuel can't go in a standard dumpster either. The City also counts painted wood, painted drywall, and insulation as garbage rather than recycling, so keep them apart from clean materials when you can.
If an item on your list gives you pause, ask your account manager before it goes in.