I absolutely would not visit any of the sights that are free - the popular ones will be awfully crowded and your time is much better spent seeing places the public is not allowed to visit except on this special weekend. It is my understanding that the weekend is exactly that, Saturday and Sunday only. Use Friday and Sunday for other sights.
I happened tell a friend about my upcoming trip to Paris that included the weekend back in 2010. She loves Paris more than I do (and it's my favorite city in the world) and got excited when I told her my dates and she told me about European Heritage Days (I was completely unaware). Some things may have changed since then. Following is my experience.
There wasn't much clarity from on-line resources. In Paris there was a list of all the sights that were taking part in the event, with their hours and special exhibits (by title only). The list was printed on newspaper and was available in every TI and many other places. It was my impression that every state-run sight and most, if not all, government (city and national) buildings were open to the public and were free.
I went to the National Assembly (equivalent of the US House of Representatives) on Saturday morning and later the Hotel de Ville (Paris city hall). On Sunday morning, the Luxembourg Palace (home of the French Senate). All three began as palaces and they are in the style of and nearly as opulent as Versailles. I had about an hour wait in line at the National Assembly (there was a small marching band in military uniforms entertaining the visitors as we waited), somewhat less at the Luxembourg and virtually none at the Hotel de Ville, which also had special weekend exhibit featuring local artisans. The only other place I remember with clarity was the official residence of the Russian Ambassador ( like this one, many foreign embassy/ambassador locations are in beautiful mansions) which had a special exhibit on the last of the Romanovs and included costumes, a dining room, and a wonderful Faberge collection. There was also a Cartier jewel collection that I have photos of. Photos were permitted at all the locations I visited except Luxembourg (but they had a beautiful coffee table book on sale for €10 and it was worth the extra weight in my suitcase - fortunately I didn't exceed the 23 Kg limit.
I think every government office was open. More than one had employees at the entrance asking passersby to come in. Most of the visitors I encountered were French, probably not Parisians. The one I missed and would put in the same category as the "big three" above is the Élysée Palace, official residence of the French President.
BTW the only museum pass worth getting is the Paris Museum Pass which is for 2, 4 or 6 consecutive days.
