Most renovation and roofing debris is fine. The limits come from Louisiana's landfill rules and from what the City's debris landfill will take.
Louisiana's construction debris landfills can't take household trash, food waste, liquids, or hazardous waste. Unprocessed tires are banned from every landfill in the state.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either.
The debris landfill in eastern New Orleans has its own longer list. It turns away furniture, appliances like refrigerators and freezers, TVs and computers, tires, and treated lumber. It also refuses batteries, motors, gas cylinders, and closed containers. If one of those items turns up, the whole load can be rejected, charged a fee, and sent somewhere else.
Mention furniture, appliances, and treated wood when you book. Your account manager can help you sort the load so it goes to the right place.