As many as you can do in one night
Weight loss is a function of calories burned, and calories burned varies based on the size and weight of the individual. If your trying to lose weight, the more calories you burn the better, so really you might as well just do as many sit ups per night as you can. After while that in itself will prove to be too easy (and therefore requiring an ungodly amount of sit ups to reach your maximum). You may wish to hold some sort of weighted object to further increase the difficulty.
Of course unless your a special type of individual sit ups will be boring pretty quick, so you might want to branch out into other exercises. Just search body weight exercises and you'll find quite the variety. Keep in mind if your trying to lose weight you want to focus on cardio. You'll also want to watch your diet, because if you don't you'll find those exercises are for nothing.