The last time the U.S. government was forced to shut down was in 1995 and 1996, before many of the affected agencies even had Websites, and the government in general did not rely on--or revolve around--technology the way it does today. When the government shuts down, so will most of its Websites and online resources.
The determination of which sites or services to keep alive falls upon the administrations of individual agencies. Each decides what is essential, and there is no specific list or mandate handed down from the White House or elsewhere.
Because the last time the federal government closed its doors was before government agencies or the general population relied so heavily on the Internet, there is no established precedent for determining which sites or services are non-essential.
The simple fact is that--like national parks or the Smithsonian museums--most government agency Websites are considered non-essential. These major agency Websites will be dark--or at least mostly dark--during a shutdown:
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