Most remodel and cleanout debris is accepted. The limits come from Illinois law and from the facilities that take your load.
The Illinois EPA says state law prohibits landfilling many common electronics, including TVs and computers. The same page says state law bans landfilling used tires. Pull those out of your load before the container is picked up.
Hazardous waste, liquids, paint, solvents, fuel, and asbestos can't go in a standard dumpster either. Batteries are a fire risk in trucks and at landfills, so keep them out as well.
DuPage County residents have a free option for many of these items. The City of Naperville runs a household hazardous waste facility at 156 Fort Hill Drive, open Saturdays and Sundays from 9 a.m. to 2 p.m., free for any Illinois resident. It takes oil-based paint, solvents, pesticides, batteries, fluorescent bulbs, and propane tanks up to 20 pounds. It doesn't take latex paint or tires.
If you're not sure about something, check with your account manager before it goes in.