In reply to HeMa:
There may be maps with the hunting schedule at some areas like Franchard and Cuvier but you certainly won't find them everywhere (and they're in French). You can download this map here: http://www.onf.fr/ile-de-france_nord_ouest/sommaire/chasse_cueillette/chass... (click on "La carte et le calendrier des zones chassées" under Massif forestier de Fontainebleau). In two weeks' time it will almost be over though (stops end of February).
Basically:
-there's hunting every Monday in both the Trois Pignons/Coquibus/peripheral areas (areas in green) and Forêt domaniale de Fontainebleau (Cuvier Franchard Apremont etc. =>areas in light blue) until 26/02.
-there's hunting every Thursday in Forêt de Fontainebleau, and some Fridays (vendredi).
-there's no hunting during school holidays in the Forêt de Fontainebleau (light blue).
-there's also fox/badger hunting (I don't know how you call it in English; you know, when you have dozens of dogs followed by pricks on horseback followed by 4WDs) on Tuesdays and Saturdays until end of March I believe, but as far as I know they don't use guns (they simply harass animals with their dogs until they're too exhausted to run, or find their burrow and literally dig them out with shovels)
-you'll always be fine if you're immediately next to a carpark, like Guichot, Bas Cuvier, Sabots... And also in areas outside the map like Eléphant, Dame Jouanne, Buthiers, Petit Bois, Gréau, etc. But not in areas on private property like Boissy, better not to go there in winter (I know some do but I wouldn't, or at the very least not on weekends).