Some metro stations are closed, at least in the morning for the military parade. (the ones right near the Champs-Elysees) I think some are closed in the evening for fireworks, also, I forget.
No regular train station is closed just because it's this holiday or any other holiday.
I'll be there that day. Tell me more about the parade.
Any suggestions on good spots for viewing?
yes : arrive early ! There are lot of people.
I personnally would prefer Chartes over Fontainbleu, I suggest the parade because it is a once a year event.
Also a lot of people enjoy big boys toys.