Framing, drywall, plaster, flooring, shingles, siding, cabinets, doors, fixtures, furniture and general cleanout debris are all normal Brooklyn Center loads. The hard limits come from hazardous waste rules and from what the county facilities will take.
Nothing hazardous goes in a container: no paint, solvents, thinners, adhesives, pesticides, pool chemicals, automotive fluids, used oil, gasoline, propane cylinders, aerosols with product still in them, fluorescent tubes or batteries of any kind. Electronics, appliances holding refrigerant, mattresses and anything containing asbestos also need their own handling rather than a corner of the load.
The county drop-off network is where people most often get caught out, so it is worth stating plainly. Hennepin County's drop-off facilities do not accept construction materials or remodeling debris, and the same list excludes concrete, asphalt, carpet, wood and asbestos. The county also says that in general it does not accept waste from businesses, with narrow exceptions for cardboard, scrap metal, recycling and organics, and recycling from a business capped at a single pickup load a week. A homeowner carrying bulbs and batteries is welcome there; a renovation load is not.
That is precisely the gap a container fills. Ask your account manager about anything unusual before it goes in, because one rejected item can hold up the whole load.