Normal renovation and cleanout material is welcome. The limits come from Missouri rules and from how the city landfill sorts what arrives.
Missouri has banned several items from landfills outright. The Department of Natural Resources lists major appliances, whole tires, oil, and lead-acid batteries. Major appliances means refrigerators, freezers, washers, dryers, stoves, ovens, dishwashers, trash compactors, and water heaters. Microwaves were taken off that list in 2002 and may be landfilled, though recycling them is still better.
Whole tires are the conditional one. They cannot be buried unless they have been cut, chipped, or shredded first, so tires have to leave your container separately.
Yard waste is kept out as well, and the city landfill composts it rather than burying it. Hazardous waste, liquids, paint, solvents, fuel, and asbestos never belong in a standard container anywhere.
The City also runs a motor oil and antifreeze drop-off with an annual limit per household. Ask your account manager about anything you are unsure of before it goes in.