As I understand it, outdoor US national parks will be open for visits during this shutdown, but at the visitor's own risk since rangers won't be around to advocate safety around wildlife. This also assumes you can access the park, since no one would be around to clear snow from roads during the shutdown. National sites inside buildings may be locked up, and sites that require transportation, like the Statue of Liberty, will not be accessible. Outdoor memorials, like those in DC, should remain open for visitors.
I'd recommend for any travelers to research beforehand whether the park or site you want to visit will be open/accessible. The good news is that for those open to visitors, there will be no one available to collect fees.
Here are some articles on how the shutdown is affecting US national parks and sites:Washington Post article on national parksLos Angeles Times article on national parksFox Business list of parks opened and closed