The National Weather Service will continue to operate despite the government shutdown that kicked in this morning. Weather.gov was functioning as of this morning and most basic operations at the National Weather Service soldier on despite Congress' failure to pass a continuing resolution (or even a budget, which it has not done since 2008) to fund the government.
There will be employees in NOAA and the NWS who are in the "don't come to work" camp but those whose responsibilities include forecasting, issuance of warnings, plus radar, satellite, and other technical support are required to work.
Within NOAA's 12,000 employees, over 5,300 employees are placed in exception status -- those whose job description fits within the "protecting life and property" camp. 3,935 of those 5,300 work for the National Weather Service. They will not get paid during the shutdown, only getting back pay upon passage of some sort of funding bill, but they will still have to show up at work and forecast weather. Technical discussions will still be written, along with forecasts, and other pertinent info. Social media usage may be a bit different, however.
Also, the GFS and NAM computer models will continue to run every six hours, just like prior to the shutdown. They will not be any better, any worse, or any different than before.
For .gov's forecasting arm, despite no pay for work for perhaps a while, things will continue "as is" and forecasts will still be issued.