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That said, police officers can write tickets. So tread carefully, and don't park in front of a fire hydrant or anywhere else where your car may be deemed a public safety hazard.
Shutting down is a scare tactic...something will come through at the midnight hour.
In regards to the originial question, yes the museums, tour buses etc would be shut down. However, just seeing the monuments would be amazing. You can walk up to them, through them, take pictures. You may not be able to get into Arlington Cemetary, but the Lincoln, Washington, WW1, WW2, Vietnam, White House area all there for you to look at and walk around outside. You could also go visit Great Falls, VA, the Kettler Complex in VA where the Washington Caps practice etc. The latter few are if you have a car. There is so much else you could do if they close, but I don't think they will. ENJOY!