Maybe nothing -- at least at first. During the 1995 and 1996 shutdowns, federal courts except for some bankruptcy courts continued operating, as the chief judges of the courts designated their employees "necessary and essential" to the continued resolution of cases. If this happens again little change in operations would take place, at least for a few weeks. But a government shutdown could affect payments to federal defenders and private lawyers assigned cases under the Criminal Justice Act. Payments would begin to be deferred once current funds are expended.
Even as courts are likely to be spared the immediate effects of a shutdown, they would be affected by the shutdown of other federal government agencies closely associated with the courts. According to the New York Law Journal, in the event of a shutdown criminal litigation would continue without interruption as an activity "essential to the safety of human life and the protection of property," but U.S. attorneys' offices could be forced "to stop or significantly curtail" other activities, including most civil litigation.
Here's the official statement from the United States Courts website:
If Congress is unable to agree on the continued funding of government before April 8th, the Judiciary is prepared to use non-appropriated fees to keep the courts running for up to two weeks.
Once that funding is exhausted, however, the federal court system faces serious disruptions. Following their own contingency plans, federal courts would limit operation to essential activities.
For the federal courts, this would mean limiting activities to those functions necessary and essential to continue the resolution of cases. All other personnel services not related to judicial functions would be suspended.
The jury system would operate as necessary, although payments to jurors would be deferred. Attorneys and essential support staff in federal defender offices and court-appointed counsel would continue to provide defense services as needed, but again, payments would be deferred. Courts would determine the number of probation office staff needed to maintain service to the courts and the safety of the community.
And here'sa message from the District Court of the Eastern District of Michigan:
During the first two weeks of a shutdown of the Federal Government, we do not anticipate any significant impact on court operations or services in the Eastern District of Michigan. The Federal Judiciary has fee generated and other non-appropriated funds that will allow it to temporarily fund all court operations. In the event a shutdown lasts longer than two weeks, court operations and services may be severely impacted. Additional information will be provided if a lengthy shutdown occurs. The Bar and public should continue to monitor the Court’s website at http://www.mied.uscourts.gov where updates will be posted as needed. To reiterate, in the event of a general shutdown of the Federal Government, there will be no immediate change in Court hours or services unless otherwise noted.