Framing, drywall, roofing, flooring, cabinets, fencing, furniture and ordinary cleanout junk are all fine to load. The limits come from California's list of material banned from the trash and from what the receiving site will accept at the gate.
California keeps hazardous material out of the garbage entirely. CalRecycle lists paint, solvents, pesticides, pool chemicals, antifreeze, motor oil, every type of battery, fluorescent and LED lamps, propane cylinders and all electronics, so televisions, monitors, computers and microwaves have to leave by another route.
Two of those catch remodels in this part of the county. Treated wood, the preserved lumber used for fences, planters and sill plates, is handled as hazardous waste in California and cannot ride in a standard container. Tires are not hazardous, but they are banned from the trash and need their own arrangement and their own charge.
Liquids of any kind, sealed drums, hot ashes and medical waste are out as well, and asbestos has to be dealt with under its own rules before demolition debris leaves the site. Los Angeles County runs a free household hazardous and electronic waste program for the material no container can take. If an item leaves you unsure, ask your account manager before it goes in.