Most household and construction debris is fine. The limits come from Texas rules that apply to every landfill in the state.
The Texas Commission on Environmental Quality lists wastes no landfill may accept. They include lead-acid car batteries, used motor oil and oil filters, whole tires, liquid waste, regulated hazardous waste, PCB waste, and radioactive material. Refrigerators, freezers, and air conditioners, including window units, need their refrigerant handled properly before disposal.
Tires have one exception. The state allows scrap tires in a permitted landfill only if they've been split, quartered, or shredded, so whole tires should go to a tire recycler instead.
Paint, solvents, fuel, and asbestos stay out of a standard dumpster as well. Ask your account manager about anything you're unsure of before loading it.