Ordinary household and construction debris is fine. The limits come from hazardous waste rules and from what the receiving facility will accept.
Hazardous waste never goes in a container. That covers liquids of any kind, paint, solvents, fuel, and asbestos, along with pesticides, pool chemicals, and cleaning products. Alexandria runs a household hazardous waste collection day for exactly that material, and the number for it is 318-441-6204.
Automotive material has its own route as well. Used motor oil, oil filters, transmission fluid, antifreeze, and car batteries all go to a local drop-off rather than into your load. The City publishes a list of places around town that take them.
Electronics should come out before pickup. Televisions, monitors, desktop computers, and printers are accepted at several drop-off points in Alexandria, and they do not belong in a mixed container.
Tires are handled separately too, and concrete is worth a word of its own. The City does not collect it at the curb, so a clean concrete load in its own container is usually the cheapest way to move it. When in doubt about an item, ask your account manager before you load it.