Labor Day in Paris

Yes, Notre Dame Cathedral is open every day of the year from 8:00 to 18:45. The tower and the crypt is not open May 1.
I would guess all the other churches are open on May 1st as well. They still hold mass on holidays.

But almost everything else will be closed, except the Eiffel Tower.

Edit - There will probably be peaceful protest marches, which will close some Metro stations. Buses will probably be discontinued during the march.
If you are concerned about the marches - called "manifestations" - you can just wander around someplace else.

Jacquemart-Andre is wonderful and will be open.

We arrived in Paris on a Labor Day a few years ago and found taxis scarce, buses on a shortened schedule, and many local shops & grocery markets closed.

Buses most likely to be affected are those that cross the axis of streets that run between Nation and Republique including Bastille.

Just Google the date and route of the manifestation and you'll see where the bus will - or will not go.

Eiffel Tower, Seine Cruises, Jacquemart Andre Museum and some other sites are open. Parks and many cafes will be open. Transport is running. Good day for a stroll and to have lined up dinner arrangements in advance.

And buy some Lilly of the Valley (muguet des bois) from one of the many people on the streets offering them on May 1.