Most construction and cleanout debris can go in your container. California's statewide rules and the county landfills set the limits.
CalRecycle lists items banned from the trash across the state. The list includes all batteries, fluorescent and LED lights, latex and oil-based paint, pool chemicals, pesticides, antifreeze, motor oil, tires, asbestos, chemically treated wood, propane tanks, and electronics like TVs, computers, printers, and microwaves.
The county landfills also turn away asbestos, munitions, explosives, and other harmful items. They do accept small amounts of some household hazardous waste from residents, such as antifreeze, used motor oil, paint, and batteries, so those can go on a separate trip.
Hazardous waste, liquids, solvents, and fuel stay out of a standard dumpster. If you aren't sure about an item, ask your account manager before loading it.