Most household and construction debris can go in your container. California's rules on what can't go in the trash set the main limits.
CalRecycle keeps a list of wastes banned from the trash. It includes all batteries, fluorescent tubes and other mercury lamps, TVs, computers, and monitors. It also covers latex and oil-based paint, aerosol cans that aren't empty, motor oil and filters, antifreeze, pesticides, propane tanks, tires, asbestos, and treated wood. None of those belong in a dumpster.
Hazardous waste, liquids, solvents, and fuel are out as well. Downey adds a local twist for bigger jobs: concrete, brick, asphalt, and soil must be recycled, so plan to keep them apart from other debris.
Check with your account manager before loading anything you're not sure about.