Ordinary construction and cleanout debris is fine. The limits come from the county facility and from Perry's own list of what the city will not handle.
Hazardous material is out everywhere: cleaning fluids, crank case and cutting oils, paints, acids, caustics, poisons, and drugs are all named in the city code as unacceptable. So are explosives, chemicals, radioactive material, oil sludge, and medical or biological waste. Liquids, septic waste, and human remains are refused as well.
Asbestos in identifiable quantities cannot go in a container. The county does accept asbestos, but at its own per-ton rate and under its own handling rules, so a project that will generate it needs to be flagged early.
Tires are specifically barred from dumpsters by city code, and the county charges a very high per-ton rate for them, so keep them out and handle them separately.
The city's own bulk collection will not touch contractor work at all. Its published refusals include auto parts, dirt, shingles, pallets, concrete blocks, concrete, bricks, fuel and oil tanks, building materials from a contractor, and remodeling debris from contractors or commercial properties. That is exactly why a container is the normal answer here. If an item has you unsure, ask your account manager before it goes in.