Ordinary Hoboken renovation and cleanout debris is fine. The limits come from New Jersey's hazardous rules and from the City's own list.
Hazardous waste, liquids, solvents, fuel, used oil, and asbestos cannot go in a standard container. Hoboken treats non-latex paint, varnish, turpentine, and pesticides as household hazardous waste with their own route through the county's twice-yearly drop-off dates, not the container.
Latex paint is a Hoboken-specific case worth knowing. An empty or fully dried can goes out with regular garbage, a half-full can can be dried out with cat litter or newspaper, and a can more than half full goes to the City's recycling site rather than into your container.
Ammunition, firearms, and anything resembling ordnance are referred to the City's public safety office and never go in a container.
Containers also have to stay covered overnight in Hoboken and cannot be filled above four inches from the rim, so overfilling is a compliance problem as well as a safety one. Your account manager will tell you if your projected volume needs a swap rather than a heap.