Most debris from a remodel, move-out, or build can go in your container. Georgia's landfill rules and the landfill that takes your load set the limits.
Georgia doesn't allow whole scrap tires in a landfill. Tires have to be shredded, chopped, or chipped first, so keep them out of your dumpster and mention them when you book.
Hazardous waste, liquids, paint, solvents, fuel, pesticides, motor oil, propane tanks, and asbestos can't go in a standard dumpster either. Batteries should be pulled out and recycled.
For leftover paint, Chatham County suggests mixing latex paint with kitty litter until it dries, then setting it out with household trash with the lid off. Unopened paint and oil-based paint have other outlets, which the county lists on its drop-off center page.
If an item seems like a gray area, ask your account manager before it goes in.