Most remodeling and cleanout debris is fine in your container. California's hazardous waste rules set the limits.
CalRecycle lists the items that can't go in the trash anywhere in the state. For a typical South El Monte job, that means batteries, fluorescent tubes, TVs, laptops, phones, paint, half-full aerosol cans, motor oil and filters, antifreeze, pesticides, propane cylinders, asbestos, and treated wood. Tires are also banned from the trash, even though they aren't hazardous.
Liquids, solvents, and fuel stay out as well. The Sanitation Districts warn that a customer who brings unacceptable waste to Puente Hills will be charged at least $500 to cover the extra cost, as their rate page states.
That's a good reason to sort before you load. Ask your account manager about any item you're not sure of.