To a large extent, it depends on what agency is responsible for the matter, (see sidebar "Who's in Charge?") and whether the matter is considered 'essential." If this shutdown follows past protocol, here is what we can expect for different matters:
Visa Applications before U.S. Consulates Abroad (Department of State)All visa processing will likely shut down except for diplomatic visas and life-or-death matters. Unless you are waiting for Humanitarian Parole for urgent medical treatment, you will probably need to reschedule any visas interviews that fall on days the government is shut down. Phone messages and e-mails about non-emergencies will not be answered, and new applications will not be processed during the shutdown.
Petitions before the U.S. Citizenship and Immigration Service (USCIS)Petitions filed by U.S. relatives and employers, green card applications, and naturalization applications will be processed as usual. So plan to keep your appointments for biometrics and any interviews. This is the upside to the high filing fees! Since USCIS is mostly funded by these filing fees rather than out of the federal budget, operations will continue despite the shutdown.
Immigration Court (Executive Office for Immigration Review)For the most part, the immigration courts will be shut down and personnel furloughed. However, there is an exception for cases involving immigrants who are in detention. Hearing for detained individuals should be considered an essential function, and will most likely be held as scheduled.
If you are not in detention, it is likely that your hearing will be rescheduled. To be safe, though, it is best to show up at your scheduled time. Better a wasted trip to the immigration court than an in absentia removal order.
Border Crossings and Ports of Entry (Customs and Border Protection)Inspectors and border patrol are considered essential personnel. However, it is possible that processing of applications at the border (such as TN visas for Canadians) will be affected.
PERM Applications and Labor Certification (Department of Labor)The Department of Labor will cease all processing of these applications if the government is shut down, and personnel will not answer inquiries by phone or e-mail.
For information on how a government shutdown may affect other parts of the government, see this article from the Washington Post.