Ordinary renovation and cleanout debris is welcome. Wood, drywall, plaster, flooring, roofing, cabinets, furniture, and general junk all load without a second thought.
New York draws a hard line at electronics. Since the start of 2015, the state has not allowed a TV, monitor, computer, printer, or game console to go in a landfill, in the trash, or out at the curb, under rules the Department of Environmental Conservation enforces. One old set buried in a full load can cost you the whole load, so pull them out before you start.
The county landfill is permitted for non-hazardous material only. That keeps out hazardous waste, liquids, wet paint, solvents, fuel, and asbestos, and the city will not collect industrial, infectious, or hazardous waste either.
Tires, appliances, and anything with refrigerant need to be handled separately and usually carry their own charge. Mention them to your account manager up front and they will sort out the right route.