Most remodeling and cleanout debris can go straight into your container. The limits come from New York State rules and from what the receiving facility will take.
New York has barred many electronics from the trash since January 1, 2015. The Department of Environmental Conservation says you can't throw away TVs, computers, monitors, printers, tablets, and game consoles with regular waste. The state also bans burying whole tires in a landfill, so tires need to stay out.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos don't belong in a standard dumpster either. The City also warns that needles, syringes, lancets, and other sharps must never go in trash or recycling, because they put workers at risk.
For electronics, the City points residents to community drop-off events. Keep TVs and computers out of the container, and mention tires and appliances when you book.
If you're unsure about an item, ask your account manager before you load it.