Most remodel and cleanout debris is welcome in your container. California's rules and the receiving facility set the exceptions.
The state's Department of Toxic Substances Control groups a set of common items as universal waste, and they can't go in the trash or into a landfill anywhere in California. That covers batteries, electronics, old TVs and monitors, fluorescent tubes, mercury devices such as thermostats, aerosol cans that still have contents, and solar panels.
Leftover paint has its own path. California runs a statewide paint recycling program that takes back old cans, so set them aside for a drop-off site rather than tossing them in the container.
Hazardous waste, liquids, solvents, fuel, and asbestos stay out of any standard dumpster. Tires and appliances need separate handling, and the transfer station charges per piece for them, with an extra charge for a refrigerator that has no proof its refrigerant was removed. Mattresses are free to drop off there in small numbers, so keep them out of the container and tell us about them.
When an item has you guessing, run it past your account manager before it goes in.