Most debris from a remodel or cleanout is fine. The limits come from New York State rules and from the City's landfill.
New York bans certain electronics from the trash. The New York State Department of Environmental Conservation says that since January 2015, covered electronics can't go to landfills or in regular trash. That includes TVs, computers, monitors, printers, and game consoles.
The Rapp Road landfill has its own list. It won't take yard waste such as grass, leaves, stumps, trunks, or limbs. It also refuses liquids, gas cylinders, barrels and drums, motors and other solid metal pieces, and chemicals or other hazardous waste. Scrap metal, appliances, tires, motor oil, and textiles are treated as recyclables and turned away as well.
As always, keep paint, solvents, fuel, and asbestos out of a standard dumpster.
If you're not sure whether something can go in, check with your account manager first.