The first reason is that building up muscle on your belly will make your belly bigger.
The second reason is that when you stop doing sit-ups, the muscle will turn to fat.
Some people start doing sit-ups because they are trying to get their belly to go away. If your belly is big because of a curve in your spine you will be effectively compressing your spine in order to make your belly look smaller. This compression leads to a bigger curve so it is a self-defeating process. (Also known as "pooching syndrome.")
Another possibility is that your belly is big because you are over-eating. If you are over-eating and you do sit-ups, the extra exercise will "tonify" your appetite causing you to want to eat more! Yum, yum. (This is often conflated with edema or bloating which can have a variety of causes, none of which are helped by sit-ups.)
Another possibility is that you do sit-ups to make your back rigid so that you won't feel a chronic injury. This sort of works but the problem is that it makes you insensitive so you are more likely to injure yourself again in the future (and more likely to over-eat).
Making one's belly and back rigid is popular with some athletes because they are always getting injured from direct impact. If two balls of equal mass collide, the denser of the two will survive and the less dense body will disperse. (This is known on the school yard as the Blamo effect!) For instance, football players often disperse (detach) their retinas this way.
The Blamo effect always works! Its physics! The denser you are the better. Unfortunately there is no art in this. The quickest way to make your body dense is to fall really hard onto a surface like ice or concrete. (A couple times a day and you'll be lookin' like Schwarzenegger in record time. Warning: This may effect your brain.)
Some people like to wear their armor on the inside. Rather than picking up some leather or even chain-mail from the local Walgreens which has the benefit of being effective against sharpened steel, they have decided that they want armor 24/7. Yes, even in the shower! These fighter-exemplars are in constant fear of a surprise attack; they need not worry about over-eating because they are too scared to eat.
There is a group of martial artists who think that making the area between the ribs and the hips rigid will give them more power. The logic here, if we can call it logic, is that a rigid body moves as a single piece and is therefore able to use its whole weight for fighting. If by fighting they mean World Wrestling Federation body-slams, than they are indeed correct. However if your idea of fighting involves mobility, and the possibility of generating explosive power from all the soft tissue in your body, tight abdominal muscles will totally break your power.
Tight muscles reduce movement range and sensitivity. They cut off the flow of power from one part of the body to another and they require constant maintenance. The more alive your whole torso is, the more power and flexibility you will have.
Future Blog: Why are/were some famous Taijiquan masters fat?