International Space Station monitoring will be maintained, but with the bare minimum of ground crew. (NASA will cease regular updates of the astronauts' activities during the furlough, although we presume if something urgent happened there would be an update.)
Robotic missions that are already in operation -- think the Cassini spacecraft circling Saturn, or the Lunar Atmosphere and Dust Environment Explorer (LADEE) winging its way to the moon -- will have small crews making sure that they are functioning properly. No scientific analysis will be conducted, though.
Certain other programs will continue if a shutdown would be detrimental to their performance. Space News reports that the much-delayed James Webb Space Telescope will be among them, as some of its instruments are undergoing cryogenic vacuum testing at the Goddard Space Flight Center.
Update, 1:09 p.m. EDT: Several missions run out of the Jet Propulsion Laboratory and Applied Physics Laboratory are running as usual for at least the next week because these facilities are running under contracted money from NASA and still have funds in the bank. According to the Planetary Society's Emily Lakdawalla: "At JPL, that includes: Curiosity; Opportunity; Odyssey; Mars Reconnaissance Orbiter; Cassini; Dawn; Juno; Spitzer; the Voyagers; and WISE, among many others. At APL, that includes MESSENGER and New Horizons. It also includes the Deep Space Network."
HiRISE is still open for business and updates will continue as usual.
HiRISE is still open for business and updates will continue as usual.