The Judiciary, which has been operating during the shutdown on court fees, will no longer have an operating budget on October 17, unless the government reopens. Coincidentally, that day is also the deadline to raise the debt ceiling (unless a proposed six-week delay passes). The Judiciary originally thought it would have to close down federal courts on the 15th, but it's been operating on a tight budget, with many civil cases postponed. If the courts run out of money, probation departments and federal public defender services will no longer be available.
Of course, each federal court has been operating a little differently during the shutdown. U.S. District Judge John D. Bates, director of the court’s Administrative Office, told the Washington Post that his office will help court executives decide what to do "after our fee balances are exhausted." It's unclear whether ongoing cases will be halted right away. Earlier this week, the D.C. Circuit Court refused a request by government lawyers to stay arguments on a Guantanamo Bay case — at issue is the practice of force feeding war-on-terror prisoners. Oral arguments on that case were set to proceed on October 18.
The U.S. Supreme Court said it would decide today whether or not it would keep operating next week, but no announcement has been made yet. The high court just opened this week, and the justices are currently hearing oral arguments for McCutcheon v. Federal Election Commission, a campaign finance case. The court is expected to hear arguments for an affirmative action case, Schuette v. Coalition to Defend Affirmative Action, next week.
Though it's looking more likely that President Obama and House Republicans will reach a short-term deal to lift the debt ceiling, neither camp is negotiating on the shutdown. And so the fate of the courts (and the people involved in ongoing cases) remains in the capable hands of Congress.