Does anyone know of impending closures of museums and other tourist sites in Rome on that day?
You can check yourself on each site’s website
Here is Colosseum- only closed on Christmas Day
https://www.coopculture.it/en/products/ticket-colosseum-forum-and-palatine-_24h/
I suspect most if not all museums are open
We have been in Rome a couple of times during the Republic Day and hardily noticed any difference. It was summer, it was hot and it was crowded. If they do any special like a parade or something,, we did not see it. It seem like any other day in Rome.
There will be a ceremony and parade and tri-color flyover at Piazza Venezia in the morning, so the Vittorio Emanuele monument will be closed while that is happening, but everything else should be open.
The military parade is held on the Viale dei Fori Imperiali next to the Forum, between the Colosseum and Piazza Venezia (Victor Emanuel Monument). So they might close access to those monuments or at least it will be crowded to reach any of them.