Lumber, drywall, roofing, tile, flooring, carpet, cabinets, fixtures, furniture and general cleanout debris are all ordinary roll-off material.
The hard limits come from the City's own code and from state law. Venice's code names what may not go out for collection in the municipal solid waste program at all: biohazardous or biological waste, lead-acid batteries, hazardous waste, incinerator ash other than crematory ash, and whole tires. Those have to be recycled or taken to a properly permitted route instead.
Beyond that list the usual exclusions apply. No liquids, no paint or solvents, no fuel or motor oil, no antifreeze, no pool chemicals, no pressurized cylinders and no asbestos. Asbestos is a live question in Venice given the age of the housing stock, and it turns up in floor tile and its adhesive, in pipe wrap, and in textured ceiling coatings. It needs a licensed abatement contractor and a separate route.
Electronics need their own handling as well. Monitors, televisions, computers and printers do not belong in a roll-off container, which matters on a condominium turnover where furniture and hardware come out of the same unit.
One Venice specific is worth flagging at booking: a container leaving a local site has to be covered in transit under the permit terms, which is a reason not to overfill above the rails.
Mention anything unusual to your account manager when you book, and the separate route for it gets arranged alongside the container.