Contrary to earlier indications, the Department of State has announced that diplomatic posts will continue to provide passport and visa operations, as well critical services to U.S. citizens overseas, despite the government shutdown. These activities are funded by user fees and are therefore not affected by the lapse in appropriations.
Caution: in the event the shutdown lasts for an extended period, funds from application fees may be depleted. The State Department could then suspend visa processing or limit it to urgent medical or humanitarian cases only. Further, applications subject to a security clearance could experience additional delays since many government agencies take part in the clearances process in addition to the State Department, and some may be affected by the shutdown.
As always, it is prudent to check the latest information provided by the specific Consulate you plan to visit.