If the container sits entirely on private property, such as your own driveway or a fenced job site, you will not need a City permit in Birmingham. That changes as soon as any part of it occupies a street.
The City code is direct about it. Nobody may occupy a street with materials or machinery tied to the construction, demolition, or repair of a nearby building, or for any other purpose, without a permit from the city engineer, a cash deposit, and an insurance policy on file. A second provision bars placing any article or obstruction in a street outside the permitted route, and the few exceptions it lists, such as briefly moving things in and out of a building, do not cover a container left in place for a week.
The permit comes from the Engineering Department. Call the city engineer's office at 248-530-1850 before you book a delivery date that depends on street space. The City Hall switchboard, at 248-530-1800, can point you to the right desk if the line is busy.
The City sets the permit fee, the deposit, and the insurance limits in a separate schedule rather than in the code itself, and it does not publish a review time, so treat the paperwork as the long pole and start it early. Unspent deposit money comes back after the work is finished. If you are not sure whether your drop spot counts as street, your account manager can help you find the right contact.