Most remodeling and cleanout debris can be loaded, but Massachusetts is stricter than most states about what may be thrown away, and the rules land on the hauler as well as on you.
The state's waste disposal bans keep a long list out of landfills and transfer stations. The ones that matter on a job site are asphalt pavement, brick and concrete, clean gypsum wallboard, ferrous and non-ferrous metals, treated and untreated wood, cardboard, leaves and yard waste, whole tires, large appliances, mattresses, and textiles. Those materials have to be recycled or recovered rather than buried.
That does not mean they cannot go in your container. It means the load gets sorted, and separating the obvious material up front usually costs less. Talk to your account manager about which materials to keep apart before you start loading.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos stay out of a standard dumpster entirely. Lead acid batteries and older television and monitor tubes are also banned from disposal statewide, so set them aside and ask your account manager where they should go.