Lumber, drywall, shingles, flooring, cabinets, siding, and general cleanout junk all load fine. The limits here come from Indiana's rules about where waste may go, and from the city's own storage rules.
Indiana requires waste to go to a permitted disposal facility authorized to accept that type of waste, and open dumping and open burning are both illegal statewide. In practice that means hazardous waste, liquids, paint, solvents, fuel, used oil, asbestos, and medical waste all have to be handled separately and declared before anything is loaded, because the site taking your construction debris is not authorized for them.
Gary adds its own storage rules. Containers have to be watertight, rodent proof, and kept covered except while being filled or emptied, and food waste and anything rats could feed on has to stay in a sealed container until collection. Salvage and lumber left stacked on a site has to sit on racks at least 18 inches off the ground.
One local trap is worth knowing. The city sets out its own containers in neighborhoods it designates as cleanup areas, and contractor and pickup truck loads are barred from going into them. If you are working in one of those neighborhoods, a rented container is the only legal route. Ask your account manager about anything you are unsure of before it goes in.