Most household and building debris is fine. The limits come from Illinois rules and from what the landfill will take at the gate.
The city's own list is the shortest way to say it. Chemicals, medications, oils, gas, paint, hazardous materials, explosives, and toxic materials may not go in any receptacle of any kind, and have to reach a proper hazardous waste facility instead.
Tires are their own category in Illinois. The state regulates everyone who generates, hauls, processes, or uses waste tires, and funds the program through a $2.50 per tire fee collected at retail. Tires don't belong in a mixed container, so set them aside and tell us about them.
The Illinois Environmental Protection Agency also separates hazardous waste facilities from ordinary landfills, transfer stations, and compost sites. Anything hazardous has to go to a facility permitted for it, which is why asbestos, solvents, fuel, and liquids stay out of a standard container.
One useful note on what is allowed. Granite City's own definition of refuse specifically covers earth and waste from building operations, along with lumber, dirt, small amounts of rock, and broken concrete. That's the everyday remodel and tear-off material a container is built for. If one item has you unsure, ask your account manager before it goes in.