Most household and construction debris is fine to load. The limits come from Texas rules and the landfill that takes the load.
The Texas Commission on Environmental Quality bars a list of prohibited wastes from every municipal solid waste facility in the state. It includes car batteries, used motor oil and oil filters, whole tires, refrigerators, freezers, and air conditioners that contain refrigerant, liquid waste, regulated hazardous waste, PCB waste, and radioactive material.
Tires are allowed only after they've been split, quartered, or shredded, so keep whole tires out of your container. Paint, solvents, pesticides, fuel, and asbestos stay out too.
Even the City's free landfill days turn away appliances with Freon, containers holding liquid, and hazardous items like pesticides, car batteries, motor oil, and medical waste. The same limits apply to anything you load in a rented container.
If you're unsure about an item, check with your account manager before you toss it.