Depending on the type of records you are requesting to have sealed, there may be a hearing scheduled after your paperwork is filed. See this page from Mass Legal Help for the different types of request.

The Legal Advocacy and Resource Center has phone numbers for questions and may have some experience in how long such requests take.
The Administrative Office of the Commonwealth of Massachusettes District Court has a detailed Guide to Public Access, Sealing and Expungement of District Court Records which describes the process and criteria in detail, but does not give any time frames for the process.