Ordinary building debris is fine: framing, drywall, plaster, siding, roofing, flooring, cabinets, decking, carpet and general cleanout material.
California keeps a long list out of the trash entirely. Paint, solvents, used motor oil and filters, batteries of every kind, fluorescent tubes and other mercury lamps, and electronics all need separate handling. CalRecycle publishes the statewide list of material banned from the trash, and the duty sits on whoever generates the waste.
El Cerrito writes the same point into its own rules. The city code states that universal wastes may not be landfilled under state law, and it separately forbids putting hazardous or household hazardous waste into any container or delivering it to a landfill, transfer station or recovery facility that is not permitted to handle it. Sharps and unwanted medicines are kept out of every container too.
Asbestos, free liquids and contaminated soil are also excluded. El Cerrito's housing stock is old enough that asbestos shows up in old floor tile and pipe lagging, so testing before demolition is time well spent.
Anything you are unsure about is worth one question to your account manager before it goes in the container.