May 1 - Siena

Government office, schools, banks will be closed. Large stores will be open, some small stores will be closed. However, restaurants, bars, and visitor-oriented stores will all be open. Public transport will be on Sunday (festa) schedules. Hotel reservations months in advanced are advised for popular areas such as Siena. You'll have a normal visit and the chances you encounter some enjoyablr special local event are improved.

At least some restaurants and coffee bars will be open, but not all.
Museums will be closed for sure.
I doubt any shops will be open, except some shops catering to tourists (but I doubt it).
I doubt also the SITA buses will be operating at all. You might need to take a train.
Most events on that day are related to rallies organized by labor unions, with speeches in major piazze, and/or workers' marches.
You will see a lot of red flags, some with hammer and sickle or Lenin's portrait, but don't worry, it's not the Red Army invading from the Soviet Union.

I'm sure Roberto da Firenze is right about the museums. The store openings are more difficult to predict as the law changed a few years ago but traditions remain. The Galleria PortaSiena shopping mall is open, for example, but individual stores will decide for themselves.