Ordinary renovation debris is exactly what these containers are for. The limits come from city rules about separate handling and from what each receiving facility is authorized to take.
New York City requires several materials to be kept out of a general load and collected separately. Those materials include fill material, medical waste, electronic waste, textiles, landscaper yard waste, grease, and paper collected for shredding. Hazardous waste, liquids, paint, solvents, fuel and asbestos never belong in a container, and asbestos is a live issue in prewar Brooklyn buildings.
The receiving facilities add their own limits. New York State authorizes each Brooklyn site for named waste types, and several of them take only clean concrete, rock, brick and asphalt. A mixed renovation load and a clean masonry load go to different places, which is one reason keeping them separate can cost you less.
One more rule worth stating plainly: the city says it is illegal to dump waste or debris on any street, lot, park or other property, and illegal to remove or dispose of trade waste without the right city registration. That is why a licensed container is the simple answer here.
If you are not sure about an item, ask your account manager before it goes in the container.