Just after la rentrée (end of vacation/back to school) in mid-September the French have an annual event called les Journées du Patrimoine. This are days when many monuments, churches, public buildings and other sites of historical and cultural significance which are normally closed to visitors are thrown open to the public.
On these days the more popular places (the Palais de l’Elysée (the French President’s residence), the Sénat and others) tend to be packed and have long lines of people waiting to get in. This site has some suggestions for places that are just as good but a little off the beaten path. Paris Weekends also mentions that the idea has spread around Europe with many member-states holding their own Heritage Days. Here is a list by country.
If you just happen to be in Paris this weekend, check out the Mayor of Paris website which has a list of events and even an Ipad/Iphone application called Patrimap that you can download as your guide. I've gone to Paris in past years to see the sights and have never ever EVER been disappointed. It's a wonderful idea and it is the most concrete manifestation of the notion that cultural heritage is not something to be hidden away and protected at all costs, but something to be shared and appreciated by all.