Most federal courts are taking preliminary steps to curtail services during this second week of the government shutdown, but that may change after October 15, as the Center for American Progress reports :
The U.S. courts posted a statement  before the shutdown began announcing that “the federal Judiciary will remain open for business,” but only “for approximately 10 business days.” The courts’ status will be reassessed “on or around October 15, 2013,” the statement promises. A message on the U.S. Supreme Court’s website  notes that the high court “will continue to conduct its normal operations through October 11,” and will provide a further update should “the lapse of appropriations” continue past that date.
In North Carolina, the Eastern District issued this order  yesterday, saying that in the event the shutdown continues beyond October 15, courts will remain open unless otherwise notified but that travel and training will be suspended.
The Middle District has a similar notice  on its website.