Most building debris and household junk can go right in. The limits come from Texas landfill rules and from the landfill that takes your load.
The Texas Commission on Environmental Quality keeps several items out of every landfill in the state. That list covers car batteries, used motor oil and oil filters from home oil changes, whole tires, liquid waste, and hazardous waste. Fridges, freezers, and air conditioners that still hold refrigerant are banned too.
Paint, solvents, fuel, and asbestos also stay out of a standard dumpster.
The City's own bulk pickup turns away TVs, tires, propane tanks, and any hazardous material, so those items need their own plan whether you rent a dumpster or not. Mattresses are taken at the county landfill for an added fee, so mention any you plan to toss.
Not sure about something? Ask your account manager before it goes in.