The commission said any work that's required for "the protection of life and property" will press on and since spectrum auctions are funded by auction proceeds, they'll continue as well. On Wednesday, the FCC plans to release a Public Notice that will describe what effects the shutdown will have on commission services like electronic filing and database systems, filing deadlines, regulatory and application fee payments, and transaction shot clocks, among others.
As the funding lapse became imminent, it wasn't clear when or if the FCC would suspend its operations as that's determined by the extent to which prior year funds are available. When those funds are unavailable or are used up during the course of a funding lapse, the FCC's policy is to furlough employees and begin a shutdown. The commission said on December 22nd that it would remain open at least through January 2nd. You can read the commission's full orderly shutdown plans here and Wednesday's notice will be available on the FCC's website.