Most remodeling and cleanout debris is fine. Massachusetts waste bans and City rules set the limits.
The City's trash page explains that the Massachusetts Department of Environmental Protection bans many materials from landfills and incinerators. That list includes mattresses, textiles, old tube TVs, items containing Freon, lead-acid batteries, and whole tires. It also bans asphalt, brick, concrete, wood, clean drywall, and metal, which must go to recycling instead.
That doesn't mean your container can't hold construction debris. It means the facility that takes your load has to sort out the banned materials for recycling. Keep mattresses, TVs, tires, and batteries out, and tell us about heavy loads of brick or concrete.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either. If you're not sure about something, ask your account manager before you load it.