This can happen. Certainly it is more common with extremely long thin umbilical stalks asscoaited with the morbidly obese. Also those pateints who have had multiple hernia repairs are at high risk for complications with the umbilical stalk during abdominoplasty prcedures.
When you do a tummy tuck the tissue around the belly button is tightened as part of the muscle wall plication. The amount of tightening is a judgement issue of trying to get the best results. The time I have seen the most risk to the belly button is when a hernia in the belly button needs to be repaired at the time of the tummy tuck.Any experienced surgeon will tell you there is a risk of belly button loss, however, even if the original belly button dies a similar structure can be created.Best Wishes.