Most building and cleanout debris is fine to load. The limits come from North Carolina's landfill bans and the county landfill's own rules.
North Carolina keeps a long list of items out of every landfill. It includes motor oil and oil filters, yard waste, appliances, antifreeze, whole tires, and lead-acid and lithium-ion batteries. Computers, TVs, wooden pallets, and plastic drink bottles are on the list too. The county landfill has recycling areas for many of these, but they can't be buried with your load.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos aren't allowed in a standard dumpster either. The New Hanover County Landfill no longer accepts asbestos at all.
Yard waste deserves a note in a city with this many trees. Branches and leaves should be kept apart from building debris so they can be handled separately.
If you're not sure about an item, ask your account manager before it goes in.