The European Heritage Days (les Journées européennes du patrimoine) are Sept. 15-16 this year (3rd weekend of September). Decide if you want to be in Paris then to enjoy it or if it's better to avoid Paris on those two days.
The major plus is that just about all government offices in Paris will be open to the public. To give you an idea, the official residence of the president, the Assemblee Nationale (lower house) and Senat (upper house), and the Paris City Hall are among them. These are all fabulous palaces (similar in style and elegance to Versailles) and are generally off-limits. Also all the state-owned museums are free. The major minus is that there will be lines to enter some of the venues, and Paris will be extra crowded with visitors. If this is your first time in Paris, you may want a calmer visit. If it's not, then this is a terrific opportunity.
This weekend is celebrated throughout France, so you are likely to find special openings, free days, and special exhibits wherever you are.
I didn't get to Annecy, but I loved Alsace. Will you have a car. Burgundy and Alsace are better with a car so you can explore the castles, abbeys, churches and tiny villages.