Most construction and household debris is fine. The limits come from Mississippi's waste regulations and from what the receiving site holds a permit for.
Mississippi keeps a firm list out of a permitted debris site: household garbage and food waste, liquids, sludges and contaminated soils, paint and paint buckets, oil and chemical containers, engines and motors, whole tires, all types of batteries, toxic and hazardous waste, regulated asbestos, and medical waste. Bulk fabric and paper loads, refrigerators, air conditioners, and cut or shredded tires need the state agency's approval before they are accepted.
That list is why sorting matters so much locally. A clean load of building material is straightforward; the same load with a few paint cans or a tire in it is not.
The City's code also pushes industrial and hazardous waste back to whoever generated it, to be handled by methods the sanitation superintendent approves, and it defines junked appliances and abandoned vehicles as separate problems with separate answers.
And the City's own crews will not take loose trash from a container, construction material inside a limb pile, or a felled tree. Those are the owner's or the contractor's responsibility.
When something is borderline, ask your account manager before it goes in. In Mississippi an unapproved item can cost you the whole load.