Paris on Bastille Day

There are several HOHO bus tours in Paris - l'Open Tour, Foxity, etc.
You should check to see where the stop most convenient to you will be. Buy tickets from the bus driver.
Also plan some extra time in case you miss the bus, if you decide to get off and do some sightseeing.
You will pass the Eiffel Tower and Notre Dame, but might not have time to go up the Eiffel Tower. You would have time to go inside Notre Dame.
Make sure to be at the airport 2 - 3 hours before departing for Rome.
Bastille day mostly involves a morning parade in the Champs-Elysees, then a concert and fireworks around midnight at the Eiffel Tower
The websites should indicate any possible detours. Otherwise it's business as usual, though a few restaurants and stores might be closed.

The HO-HO buses will be rerouted for hours, maybe all day, as the place de la Concorde is completely taken over by viewing stands and set-up for the parade, and the parade takes place on the whole length of the Champs-Elysées. The other issue is that it's the streets around this whole area -- even over to the Left Bank! -- that serve as the staging and retreating grounds for all the military units that are involved in the parade. It's kind of cool to see when these units are coming back the other way, but it does definitely mess up road traffic in central Paris.

You could easily, however, take the metro to say Trocadero for a great view of the Eiffel Tower, and then walk down through the gardens over to the Tower itself; you can definitely visit Notre Dame, etc. etc. (walk from Hôtel de Ville metro). I just think that it's one of the very days when HO-HO is not going to work very well.

p.s. as for public transportation, several of the stations from Concorde to Charles de Gaulle Etoile on the line 1 will be closed for a certain period during the morning. And sometimes just the EXITS are closed, so you can still transfer underground from one line to the other (e.g. from the 1 to the 6 at CDG-Etoile to go to Trocadero).

In answer to another poster's question, I had looked up and seen that the Eiffel Tower is closed a good part of the day on Bastille Day due to the fireworks prep. Plus there will be a ton of people around securing space for the concert/fireworks that evening. So, as Chexbres said, going up is probably not very likely. She's still amazing to see, though, and that you can do with no problem!

Thank you for all the feedback!