Most household and construction debris is fine. The limits here come from the parish landfill that takes your load, and its rules are stricter than many.
Without written approval from the parish's Director of Environmental Services, the North Landfill won't accept batteries, unclean or uncrushed metal drums, crumbly asbestos, medical waste, industrial waste, paints, paint thinners, liquids, solid chemicals, or incinerator ash. It also turns away hazardous waste and materials contaminated with PCBs or asbestos.
That lines up with the usual rules for any dumpster: no hazardous waste, liquids, paint, solvents, fuel, or asbestos.
A few items have their own conditions. The landfill doesn't take commercial or industrial tires, and residents can bring up to four passenger tires a year, off the rim. Refrigerators and freezers must be emptied, with the doors off and all refrigerant removed. Mention tires and appliances when you book so they're handled the right way.
Your account manager can confirm any item you aren't sure about before it goes in the container.