Most remodeling, roofing, and cleanout debris is fine. The limits come from Texas landfill rules and from the landfill that receives your load.
The Texas Commission on Environmental Quality bars several wastes from every municipal landfill in the state. They include lead-acid batteries, used motor oil and oil filters, whole tires, refrigerators and air conditioners with refrigerant, liquid waste, regulated hazardous waste, PCB waste, and radioactive material. Tires can go to a permitted landfill only after they've been split, quartered, or shredded.
Hazardous waste, paint, solvents, fuel, and asbestos can't go in a standard dumpster either. The City's own bulk pickup rules turn away household hazardous waste, tires, commercial debris, contractor debris, and Freon appliances without a certification tag.
Check with your account manager about anything you're unsure of before it goes in the container.