Most household and construction debris is fine to load. The limits come from the state disposal bans and from what the receiving facility will take.
Massachusetts bars a long list of materials from being thrown away or sent for disposal, and the duty falls on the hauler as well as on you. MassDEP's banned list covers asphalt pavement, brick and concrete, clean gypsum wallboard, ferrous and non-ferrous metals, glass, metal and plastic containers, lead acid batteries, leaves and yard waste, mattresses, recyclable paper and cardboard, textiles, white goods, and cathode ray tube televisions and monitors. Treated and untreated wood and whole tires are barred from landfills.
That does not mean those materials cannot leave your site in a container. It means the load gets processed and separated instead of buried, which is why clean single-material loads of concrete or brick often cost less to move than mixed ones.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos stay out of a standard container in every case. Braintree holds its own household hazardous waste events for small quantities from homes.
If you are not sure where an item belongs, ask your account manager before it goes in.