Most remodeling and cleanout debris can go in a container. The limits come from California's waste rules and from the sites that receive the load.
California treats several common items as universal waste, which can't go in the regular trash. The Department of Toxic Substances Control lists batteries, electronic waste, old tube TVs and monitors, lamps, mercury waste, non-empty aerosol cans, and solar panels. If a reroof includes taking down a solar array, those panels need their own handling.
The state's organic waste law also asks every city to collect food scraps and yard trimmings separately. Keep large amounts of green waste out of a construction load unless your account manager says otherwise.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos aren't allowed in a standard dumpster. Los Angeles County holds hazardous waste collection events most Saturdays, and the City lists 1-888-CLEAN-LA for dates and locations.
If you're not sure about an item, ask your account manager before it goes in.