On private property, including your own driveway or a fenced job site, Lakewood asks for no container permit. The City does license portable storage containers, but that rule is written for a fully enclosed, box-like unit, so it does not reach an open-top roll-off.
The street is a different matter, and Lakewood's rule is specific and old. Placing material for building purposes on any street, lane, alley, or public ground requires a written permit from the Director of Public Works. The permit names the part of the sidewalk and street you may use and the period of use, which may not run beyond sixty days, and in no case may you take more than half the sidewalk or half the street.
There is a night-time duty attached as well. Anything left in a street, alley, or sidewalk has to be marked with red or amber lights from dusk until daylight for every night it stays there. On a Lakewood street that is a real operational requirement, not a formality.
One rule works for you rather than against you. A construction or demolition contractor is required to provide on-site receptacles, bulk containers, or detachable units for loose debris, paper, building material waste, and scrap, with everything containerized by the end of each day and the number of containers set by the size of the job. Failing to do it is a minor misdemeanour, so the container is the compliant answer.
The City of Lakewood answers on 216-521-7580, and the Division of Refuse and Recycling takes disposal questions on 216-252-4322. Your account manager can confirm whether your drop spot needs the public works permit.