According to published reports and briefings, USCIS and ICE will continue normal operations. USCIS is user-fee based and therefore immune from the budget, while ICE is an essential function. The Department of State is also user-fee based and will continue adjudicating immigrant visas. Lastly, the immigration courts will only hear detained cases.
The Department of Labor–for those of you with employment-based cases–is apparently shutting down its immigration-related functions.