Nearly all remodel and cleanout material belongs in a container. The exceptions trace back to Illinois law and to what the transfer station will unload.
Illinois has banned landscape waste from landfills since 1990, so grass, leaves, and brush need a compost route rather than a spot in your container. Used tires were banned four years later. Large appliances, meaning refrigerators, freezers, ranges, water heaters, air conditioners, and humidifiers, stay out unless their components have been stripped first.
Electronics are out as well. State law blocks landfilling of televisions, computers, and many similar devices, and loose batteries are a genuine fire risk in trucks and sorting lines, so both need a drop-off instead.
The transfer station serving this area draws a hard line of its own. No hazardous waste, no household chemicals, no liquids, and no paint come off a truck there, and landscape waste from residents is refused outright.
One more thing worth knowing: under Illinois rules the person who generated the waste can be held responsible if a hauler dumps it illegally, which is exactly why a licensed company and a paper trail matter. Run anything questionable past your account manager before it lands in the container.