Not much. According to Hannity, 80% of DHS employees will continue working as normal. Of course Obama will probably use the old “lay off cops and teachers first” strategy, and shut down the border patrol.
It's not quite as easy as you imply. Obama doesn't get to choose who are the "essential" employees. The bureaucracy has already designated who they are and any deviation would spark a judicial mess. Since, it's not a general government shut down, he can't close the National Parks and the WWII Monument, so he will need to search for creative ways to screw with conservatives. My guess, no background check approvals for gun purchases. I think that move might backfire, so I hope that he goes that route.