Most house and job-site debris is welcome. The limits come from Missouri law and from what each disposal site will take.
Missouri keeps four things out of landfills statewide: major appliances, whole tires, oil, and lead-acid batteries. The state's list of banned items counts refrigerators, freezers, washers, dryers, stoves, ovens, dishwashers, trash compactors, and water heaters as major appliances. Microwaves came off that list in 2002 and can go in a regular load. Tire shops have to take back the tires they replace, so that's the easy route for those.
Hazardous material is out as well: paint, cleaners, drain openers, pool chemicals, pesticides, fertilizer, antifreeze, gasoline, propane tanks, and sealed drums. Anything with asbestos in it needs a licensed abatement contractor, not a container.
Televisions and computer monitors are their own problem. The city turns them away at its drop-off site, and you should pull them out before the container arrives.
Got something you can't place on that list? Check with your account manager before you load it.