On private property, such as your driveway or your job site, no city permit is needed for the container itself. That is where it has to go, and Burbank is stricter about this than most towns nearby.
The city does not issue a street placement permit, because its code does not allow one. Refuse containers have to be kept on private property, screened from view as far as is practical, and the code states plainly that no refuse container may sit on a public street, roadway or sidewalk. There is no fee that buys an exception, so plan the drop spot on your own land from the start.
The work itself is a different question. The Building Department requires a permit for demolition, roof replacement, driveway work, siding, masonry and any kind of remodeling, and the city expects every contractor working in Burbank to be licensed. If your project needs a permit, line that up alongside the container.
The Building Department is at City Hall, 6530 W. 79th Street, and the number is 708-599-5500, option 2. Public Works is at 5700 W 75th Place on 708-924-7390, open weekdays from 9 a.m. to 4:30 p.m. If you are not sure whether your spot counts as private property, your account manager can help you find the right contact.