Most household and remodeling debris can go in the container. The limits come from Massachusetts disposal bans and from the facility that receives your load.
The Town's trash and recycling page lists the items Massachusetts bans from solid waste disposal. That list includes leaves and yard waste, major appliances, tires, car batteries, TVs, and computer monitors. It also covers wood, metal, brick, concrete, asphalt, and cardboard, which processing facilities pull out for recycling.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster at all.
Keep appliances, tires, and electronics out of your container. If you have something you're unsure about, ask your account manager before it goes in.