Everyday renovation and cleanout debris is fine. The restrictions come from New York State law and from New York City's own handling rules.
Electronics top the list. New York State banned consumers from putting covered electronics in landfills, in the trash, or at the curb back in 2015, as the state's electronic waste rules explain. Televisions, monitors, laptops, and desktops have to come out before the container is hauled.
Hazardous material stays out too: liquids, fuel, solvents, paint, and asbestos have no place in a container. The city runs a special waste drop-off on Staten Island, off the Muldoon Avenue exit of the southbound West Shore Expressway, that takes paint, motor oil, batteries, fluorescent bulbs, aerosols, and passenger car tires from residents. It is open to households only, not to businesses or commercial vehicles.
Batteries deserve a separate mention, since lithium and rechargeable types cause fires in collection trucks. Keep them out of the container entirely.
Tires, appliances with a cooling unit, and mattresses are usually billed per piece rather than by weight. Flag them when you book, and ask your account manager about anything you are unsure of.