Peritonitis is an inflammation of the peritoneum, which lines both the abdominal wall and the abdominal organs. Normally this is not something that should ever be affected by body piercing, but piercings which actually pass into the inner body cavity such as anal-to-vaginal piercings and some deeper variations of anal piercing can actually do so. If a piercing such as these becomes infected it can also lead to an intra-abdominal abscess, where pus nodules build up in the area, which will further complicate the peritonitis.
In very rare cases navel piercing infections can travel inward and cause this problem (this is far, far, far more likely with true navel piercings rather than ones through the rim which can not normally have their infections pass through the peritoneum) — but I must emphasize that this is extremely unlikely.
Symptoms of this problem of course include abdominal pain, tenderness and distention, as well as a fever and difficulty urinating. You may also experience nausea, thirst and other side-effects. This is the sort of problem you should take very seriously. If you deal with it promptly (by calling your doctor or visiting the hospital), antibiotics will usually deal with it fairly quickly and easily (although you'll be forced or strongly advised to remove the modification to blame). However, if you don't seek medical attention, you could die or develop other serious problems. If you have an infection in this type of piercing, taking out the jewelry and assuming the problem will solve itself could be the death of you, as the infection becomes trapped inside your body.
Whenever a piercing or other modification has the potential to draw an infection into the inner body, the severity of problems it can cause skyrocket. Such invasive procedures are not recommended to anyone not having an excellent source of communication with one's body.