Paris on Labor Day (1 May)

Probably a parade or a demonstration. Some business will close, lots will reamin open. Some museums close (eg, Louvre), the ET is open. Nobody travels for miles to see what's going on. You can work around it or even not notice it.

As Ed said, you'll be able to do many of the touristic things. However, the parade will probably be huge on May 1st, as it falls between the first round of the presidential election and the run-off two weeks later. Go watch. This is true cultural experience.

Paris is exceptionally crowded over the entire Labor Day holiday (people usually take more than one day off!). After we saw the lines waiting to get in the Orsay (even for those who already had tickets), we skipped it since we had been on another trip. It was easier to get into the Louvre, but extremely crowded. Notre Dame Plaza was packed. I thought I had seen Paris crowded before, but the Labor Day holiday(s) were something incredible in terms of the sheer number of people. It seemed like everyone who lived in France came to Paris for the holiday weekend that year -- so it was packed for 4 solid days. If you must be there during that period, I would certainly make reservations as early as possible, especially for hotels and special dinners. Unfortunately I think most sightseeing will just be frustrating because of the number of people.
Some restaurants also close and take their own holiday during this time.