I did a quick search on Autoeurope.com for a rental car in Italy, and its says that their offices are closed on Wed., June 2. Does anyone know if this is a holiday?

YES it is called REPUBLIC DAY

Never heard of it! (then again, I'm not Italian).

Do you know what kind of closures we should expect on that day? Will the major attractions in Rome be closed, for example? Will public transporation schedules be reduced?

If I am not wrong in 2001 the present Italian president (Ciampi) set back this holiday day after 24 years that it was not duly celebrated (there was just the usual parade, but not a real holiday).
We celebrate the day the Italian Repubblic was born (June 2 1946) after a public referendum.

ooops I forgot! You night expect to find some closed museums. Also, being a holiday day the means of transportation will be reduced. And, as there is a parade in Piazza Venezia and Via dei Fori Imperiali (from Piazza Venezia and the Vittoriano to the Colusseum), so the area will be closed, I think also for security reasons (the president attends the parade and with his many ministers and members of the Army).

I was there once on that holiday and saw some fighter jets overhead. Otherwise, everything seemed open.

Thanks for all the info. I'm glad I became aware of the holiday before I left so I can plan accordingly. It pays to do your research!