Most construction and household debris is fine. The limits come from the state disposal bans, from the city's recycling rules, and from what the receiving facility will take.
Massachusetts bans a long list of materials from being thrown away or sent for disposal, and the duty sits with the hauler as well as the generator. 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 tube televisions and monitors. Wood and whole tires are barred from landfills.
Those materials can still leave in a container, because the load is sorted at a processing facility rather than buried. Cambridge goes a step further and asks for proof of recycling as part of the dumpster plan, so keeping cardboard, metal, and clean wood separated is worth doing on paper as well as in practice.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos never go in a standard container. Asbestos in particular needs a licensed abatement route before any demolition starts.
Ask your account manager about anything you are unsure of before it goes in.