Most construction debris and household junk is fine. The main limits come from Texas rules that apply to every landfill in the state.
The Texas Commission on Environmental Quality bars landfills from taking car batteries, used motor oil and oil filters from home oil changes, whole tires, liquid waste, and hazardous waste. Refrigerators, freezers, and air conditioners with refrigerant are also on that list.
Paint, solvents, fuel, and asbestos can't go in a standard dumpster either.
The City's monthly bulk pickup won't take construction debris, tires, batteries, or propane tanks, and it caps each pickup at four cubic yards. For a remodel, a dumpster is the simpler choice.
If you're unsure about an item, check with your account manager before you load it.