Ordinary building and cleanout debris is welcome. The limits come from state law and from what the county will take at the gate.
Florida bans batteries from the trash outright. The state's Department of Environmental Protection says so plainly: it is illegal to discard rechargeable batteries, or products that contain them, in the trash, and that applies to households and businesses alike. Automotive lead-acid batteries carry their own separate ban.
Hazardous waste, paint, oil, chemicals, solvents, fuel, and asbestos stay out of any container. The city will not even collect those at the curb, and neither will the county drop-off centers.
The county adds limits of its own. Its neighborhood drop-off centers take residential waste only and no commercial waste at all, and construction and demolition debris is accepted only at the county central facility, for a charge. Those centers also turn away appliances, mattresses, and oversized yard waste.
One local quirk worth knowing: the city will collect some construction material at the curb, but only if it is cut into three-foot sections, and it will not take roofing material at all. For a roof job or anything past a few pieces, a container is the only practical route. Check anything unusual with your account manager first.