One reason is what a flat stomach does to the lumbar spine. Your lumbar spine (lower back) should have a nice smooth curve to it. The abdominal muscle tension necessary to create a flat stomach takes the natural curve out of your lower back and create more lumbar flexion than you functionally want. Losing your natural lumbar curve creates a situation where your intervertebral discs are "loose packed" which puts you at an increased risk of herniating a disc. The excessive abdominal tension means you no longer have proper "dynamic tension" or the balanced coordination between the muscles in the front and back of your body. This can lead to muscle, ligament, or tendon injury and pain.
The second reason why a flat stomach is a bad thing is it interferes with normal diaphragmatic breathing. When you breathe your stomach, lower ribs, and lower back should all expand outward. We call this East-West Breathing. The tight abdominal muscles do not allow normal unrestricted expansion of the abdominal wall and usually leads to abnormal and faulty North-South Breathing. This means your ribs, chest and shoulder will rise and fall when you breathe. This takes increased energy and decreases your functional lung volume (overuse of upper lobes of lungs and underuse of lower lobes of lungs). This leads to decreased performance and increased upper back tension often resulting in sore shoulders and tension headaches.
The third major problem with a flat stomach is that the excessive abdominal tension we have been talking about also restricts movement throughout the torso and hips. When playing sports like golf, tennis, basketball, soccer, or doing things like gardening, raking leaves, sweeping, or making your bed, we need to have smooth coordinated movement of all our joints. Tight abdominal muscles disrupt the connection between the upper and lower body and can result in injury and decreased performance.
Good posture allows for normal abdominal expansion during breathing and movement. Small children are great examples of good posture and normal functional abdominal muscles and tone. When viewed from the side children will have a slightly convex abdominal wall which is normal and good posturally.