Most household and construction debris loads without any trouble. The limits come from Arizona law and from what the receiving site will take at the gate.
Arizona bars lead-acid batteries from landfills and from incineration outright, and it also says the site itself may not knowingly accept one, so a car or boat battery has to be returned to a retailer or a recycler.
Hazardous waste, liquids, paint, solvents, fuel, motor oil, antifreeze, pool chemicals, propane tanks, and fire extinguishers all stay out of a standard container. So do televisions, computers, and other electronics.
The transfer station nearest to Queen Creek also refuses household hazardous waste, refrigerators, medical waste, and liquid waste such as oil, paint, and chemicals.
One Town rule is worth knowing even though it isn't about your container. The Town's bulk trash pickup won't take construction materials at all, including concrete, dirt, rock, gravel, drywall, pavers, wood, and brick, which is exactly why a rented container is the route for project debris. Ask your account manager about anything on your list that looks borderline.