Most construction and cleanout debris can go in your container. California's trash rules and the facility that takes your load set the limits.
CalRecycle's list of wastes banned from the trash covers a lot of common household items. That includes fluorescent tubes, all household batteries, latex and oil-based paint, paint thinners, pesticides, and pool chemicals. Motor oil, antifreeze, tires, and propane tanks are banned too.
Building materials are on the list as well. Asbestos, which shows up in some older roofing, flooring, siding, and cement, needs special handling. So does wood treated with chemical preservatives, which California manages as hazardous waste.
Electronics like TVs, monitors, and computers also stay out of the trash in California. Take them to an e-waste recycler instead.
If you're not sure whether something can go in, ask your account manager before you load it.