The latest news reports that an agreement was reached at the last moment and there will be no government shutdown.
With less than five hours to reach an agreement between deadlocked political parties, it looks increasingly likely that at least a short U.S. federal government shutdown will occur. Space News has an article on NASA's shutdown plans that contains some good news. While mission development generally will be suspended, contractors using previously obligated funds can continue their work. Both planetary missions with critical launch windows, the Mars Science Laboratory and the Juno Jupiter orbiter, are being developed by JPL, a NASA contractor. In the case of Juno, the spacecraft is being built by Lockheed Martin Space Systems, by a private contractor. JPL and Lockheed Martin also are managing the development of the GRAIL lunar obiters. Therefore, according to the article, work on these missions can proceed at least until NASA employees or NASA facilities would have to become involved. The article does not address how launch preparation work would be effected, although I believe the actual launch services are provided by contractors.
In addition, mission operations will continue for all missions currently in flight.