Most remodeling and cleanout debris is fine. California's trash rules and the facility that takes your load set the limits.
CalRecycle keeps a list of items banned from the trash, and it covers more than people expect. Every kind of battery is on it, from a car battery down to a single AA. So are fluorescent tubes, LED bulbs, and electronics like TVs, monitors, laptops, printers, and microwaves.
Tires are banned too, along with propane tanks and other gas cylinders. Wood treated with chemical preservatives, such as old fence posts and deck framing, has to be handled as hazardous waste. Asbestos, found in some older roofing, flooring, and siding, can't go in either.
Hazardous waste, liquids, paint, solvents, and fuel stay out of a standard dumpster as well. Pull these items before you load, and mention mattresses and appliances when you book.
When something doesn't fit neatly on either list, ask your account manager before it goes in.