Most household and construction debris can go in the container. The limits come from Virginia waste rules and from the facilities that take the load.
Virginia law doesn't allow used car batteries in mixed trash. They have to go back to a battery seller or a recycler.
The City's list of local disposal sites notes that the waste-to-energy plant won't take construction debris, stumps, rocks, dirt, tires, TVs or other electronics, liquids, pressurized tanks, or hazardous waste. Refrigerators, air conditioners, and dehumidifiers aren't accepted there either.
Standard exclusions apply as well. Keep paint, solvents, fuel, and asbestos out of your dumpster.
If you're unsure about something, check with your account manager before it goes in.