Most remodeling and cleanout debris can go straight in the container. The limits come from Virginia's landfill rules and from the regional landfill's own policies.
Virginia requires tires to be split, cut, or shredded before a landfill can bury them, so whole tires need to travel separately. At the regional landfill, one whole tire, one battery, or one appliance in your dumpster turns the whole load into a mixed load with a higher per-ton charge.
Hazardous waste, liquids, paint, solvents, fuel, motor oil, propane tanks, and asbestos can't go in a standard dumpster either.
Mattresses and box springs need a plan of their own. The Belman Road Convenience Center won't take them, and the regional landfill charges for them by the piece. Old refrigerators, freezers, and air conditioners hold Freon and count as appliances, so keep them out of your load.
When in doubt about an item, ask your account manager before you toss it.