Start with private property, because Framingham is not a city where the answer there is simply no. A container on your driveway or parking lot needs no City street permit, but the Board of Health's own container rules still apply, and one of them is a setback.
Those Board of Health rules say a container may not be placed within 25 feet of any public way unless written permission is obtained from the Board of Health or its agent. On many Framingham lots the whole front yard is less than 25 feet deep, so written permission is the normal route rather than the exception. The same rules require the container to be metal or other durable material, leakproof and verminproof, fitted with a suitable tight-fitting cover, kept covered at all times, and emptied and cleaned often enough to avoid odors. Responsibility for all of that sits with the property owner or the owner's agent, which means you, so it is worth getting right.
If any part of the container will occupy the street, sidewalk, curb, driveway apron, or grass berm, the City requires a Street/Sidewalk Obstruction Permit, and its own permit page names temporary dumpsters related to construction as a typical activity. The fee is $50, processing takes three to five business days, and the application goes in before you obstruct City property. Insurance requirements apply to a contractor working with the City for the first time.
Two numbers to keep handy. The Engineering and Transportation Division handles the obstruction permit on 508-532-6010. Every applicant also has to notify the Framingham Police Department of a street or sidewalk obstruction, whether or not a detail is required, and the police detail line is 508-532-5907. For the Board of Health side, the Public Health Department is at 150 Concord Street on 508-532-5470.
Your account manager will tell you which of those calls your job actually needs and find the right contact for you.