The short answer is it depends on which immigration-related branch is involved and what type of case it is. The various immigration related government entities (USCIS, ICE, Department of State, EOIR, Department of Labor, etc.) have not released much info regarding what will happen; however, a similar issue arose in 2011, and the contingency plans in place then may be instructive. The following is merely speculation and should not be relied upon as conclusive (but hopefully it helps).
USCIS: Many USCIS functions will continue, because USCIS is primarily funded by filing fees rather than appropriations. E-Verify might shut down.
Department of Labor: The DOL is funded by appropriations; therefore, many tasks will be put on hold and some workers will be furloughed. DOL will not process labor condition applications, audit responses, prevailing wage determination applications, applications for temporary or permanent employment certification. The DOL iCert Visa Portal System would be inoperable as well.
Executive Office of Immigration Review: Other than the detained docket, most EOIR functions would likely cease due to a shutdown. This was the plan in 2011.
Immigration and Customs Enforcement: ICE attorneys will continue to work on detained docket cases; however, other cases will not be considered essential and will cease during the shutdown.
Department of State: Visa processing will not continue unless it is a “life or death” matter.
Customs and Border Protection: Borders will remain open during a shutdown; however, it is not clear if applications eligible to be processed at the border will be handled during a shutdown. Inspection and law enforcement are considered essential personnel and will continue to work.