The problem is, the distance between Paris-Nice is vast and you will spend the bulk of the 2 days covering distance between the cities and have little time for touring other areas, IMO.

Better, go by air and take day trips into the Arrière-Pays behind Nice with some breathtaking hill towns and perhaps a little further astray into the Var Coast or the Gorges du Verdon.

The regions of France need a bit more time-- next year?

P.S. Unless you have a lot of time for cave (as in wine) touring, which requires at least 2-3 days, and doing the degustations, you don't need to go to Burgundy. You will find wines of Bourgogne throughout the country.