Most remodel and cleanout debris is fine. The limits come from Nebraska's landfill rules and the City's transfer station.
Nebraska law bans three things from landfills: lead-acid batteries and waste oil, discarded household appliances, and waste tires in almost any form. Because every Columbus load ends up at a landfill, those items need to stay out of your dumpster.
The City's banned materials list spells it out for the transfer station. It refuses appliances such as refrigerators, freezers, air conditioners, water heaters, washers, dryers, and stoves. It also refuses slate shingles and siding, anything longer than eight feet, burn barrels, car batteries, paint, antifreeze, pesticides, motor oil, and tires. Liquid paint can't go in, but the City notes that paint mixed with kitty litter and dried out can go out with regular trash.
Hazardous waste, liquids, solvents, fuel, and asbestos can't go in a standard dumpster either. Cut long boards and pipe to under eight feet before loading.
If you're unsure about an item, ask your account manager before it goes in.