Most building and household debris is welcome. The limits come from Fairfield's own rules and from what the receiving disposal site will take at the gate.
Fairfield's container rules name the materials to keep out: hazardous waste, infectious waste, tires, electronic waste, batteries, oil-based paints, gasoline and oil, and anything similar that is harmful to public health or the environment. Treat that as the working list for any container in the city.
Asbestos is handled separately under state rules and never belongs in a standard container. If an older Fairfield home has pipe wrap, floor tile mastic or siding you are unsure about, stop and have it tested before demolition starts.
Scrap tires have their own statewide rule. Moving more than ten of them at once requires a registration from the Ohio Environmental Protection Agency, so a pile of old tires needs its own route rather than a corner of your container.
Liquids, fuel, solvents and wet paint are out everywhere. Latex paint can be dried solid first, but oil-based product has to go to a household hazardous waste collection.
If you are holding something that does not clearly sit on one side of that line, ask your account manager before it goes in. A rejected load at the gate costs far more time than a phone call.