The agencies handling immigration matters under the Department of Homeland Security are largely unaffected by the government shutdown.
U.S. Citizenship and Immigration Services (USCIS) is open for business for all fee for service activities. This means that applications and interviews for various benefits including the green card and naturalization are moving forward.
Immigration Customs and Enforcement (ICE) is also continuing its detention and enforcement operations.
Meanwhile, the Executive Office for Immigration Review (EOIR) or Immigration Court, which is under the auspices of the Justice Department, is only handling removal cases for individuals who are in custody. Removal cases for individuals who are not detained are being continued or held in abeyance.
The Washington Post has a great round up of how other government agencies are impacted by the shutdown.