Most remodeling and cleanout debris is fine. What's off limits comes from California's waste rules and from the transfer station that takes your load.
California keeps a long list of items out of the trash. CalRecycle lists waste banned from the trash, including batteries, fluorescent tubes, paint, solvents, motor oil, antifreeze, and pesticides. Tires, propane tanks, and electronics like TVs, monitors, computers, and microwaves are on the list too.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster. Treated wood, such as old fence posts and deck lumber, has to be handled as hazardous waste in California, so keep it out of the main load.
The Fremont transfer station also turns away asbestos, ammunition, explosives, and medical waste. It hosts Alameda County's household hazardous waste drop-off, which is free for county residents.
If you have an item you're not sure about, ask your account manager before it goes in the container.