What will happen to my body if I do 100 push-ups and 100 sit-ups daily for a month?Assuming you're beginning from no exercise, you're bound to notice some changes. Although pushups are a strength exercise, doing 100-reps of anything lends itself more to endurance, than to strength or muscle gain.
Remember that muscles grow as you rest, and not as you train. By pushing through the discomfort, you are actually training your body to foster greater endurance. This means that if you're pushing yourself to failure, you'll notice the necessary gaps between sets will decrease (along with the initial soreness you experience at the beginning of the month) until you are capable of doing them in 1-2 sets.
I'd follow the directions of the other Quorans and consider more effective exercises. Further, I'd recommend picking more exercises to get a whole-body workout (every second day) or exercising a different body part each day over 6 days:
Monday - Chest
Tuesday - Back
Wednesday - Core
Thursday - Shoulders
Friday - Legs
Saturday - Arms
After 6 days, lather, rinse and repeat. You'll be: working smarter, less sore, and more likely to maintain your routine at the end of the month by avoiding the boom and bust nature of your intended approach.