Given that the government has now shut down, at least for the present time, it is not immediately clear what will happen to the cases pending immigration court.
Most of the immigration courts have closed, except for those handling emergency or detained cases. Thankfully the Department of Justice has published a helpful list for determining which courts are still open and for what purposes during the government shutdown. [Note – that site has since been taken down, but its description page still remains: http://www.justice.gov/eoir/legacy/2013/10/24/Shutdown09302013.pdf]
The notice explains that the only courts remaining open are those that handle detained cases. And the notice says the BIA is processing “emergency stay requests as well as cases where the alien is detained, including case appeals, motions, federal court remands, and bonds.”
You should still be able to file a BIA appeal or Motion to Reopen, and you should do so before your deadline runs out. Don’t assume the Board will forgive missed deadlines just because the BIA or the Immigration Court are closed.