USCIS is a fee-funded agency so it will generally continue to function when the government is shut down. The exception to this is those programs that receive non-fee funds to operate, such as E-Verify, the EB-5 Immigrant Investor Regional Center Program, and Conrad 30 J-1 doctors. Those programs may be suspended or otherwise impacted.
Department of State
Visa and passport operations are fee-funded and should not be impacted by a shut down. However, if operating funds are exhausted, consular posts will generally only handle diplomatic visas and “life or death” emergencies.
Department of Labor (DOL)
Historically, DOL services have been affected by government shut downs since many of their immigration related services are not fee-funded; however, we have seen ICERT operating (except for the recent crash) so we are not entirely sure how DOL activities are being affected.
Let’s all hope this thing does not drag on forever!