Most debris from a remodel or cleanout can go in. New York's disposal rules and the landfill's own list set the limits.
The New York State Department of Environmental Conservation says that since January 1, 2015, you can't throw away many electronics in the trash or at a landfill. That covers TVs, computers, monitors, printers, and game consoles. The Town's transfer station takes these for free.
The Albany landfill treats tires, white goods like refrigerators and washers, and electronics as unacceptable waste, and it charges a fee for each one that shows up in a load. Mattresses carry an extra charge too. Pull these items out or mention them when you book.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster. The transfer station won't take batteries or fluorescent bulbs either.
When in doubt, ask your account manager before you toss it.