The Department of Education, along with other agencies, has developed a contingency plan in the event of a government shutdown. The plan can be seen here. The other agency contingency plans, including BIA and BIE, can be seen here.
A quote from the Department of Education’s plan:
As set forth in this plan, the Department would furlough over 90 percent of its total staff level for the first week of such a lapse. During this first week, we would maintain only those excepted functions related to the discharge of the duties of Presidentially-appointed, Senate-confirmed individuals;those employees charged with the protection of life and property; and, as appropriate, the obligation, payment, and support of student financial aid as well as other authorized payments and obligations.