I understand why cardiologists and endocrinologists care about the weight of their patients.  Why is this important to gastroenterologists?

One answer is fatty liver disease.  The most rapidly increasing liver disease in the American population is hepatic steatosis (excessive fat in the liver).   When fat inflames and damages the liver, it is called steatohepatitis.  When excess fat is stored in the liver and the person does not drink alcohol, the condition is called Nonalcohol Fatty Liver Disease. (NAFLD).  When NAFLD is associated with liver cell inflammation in a person who does not drink alcohol, the condition is called NASH (Nonalcohol Steatohepatitis). Fat can be toxic to the liver, clogging the liver cells and preventing them from functioning properly.  Fat can inflame the liver in some circumstances and increase the risk of developing cirrhosis and liver failure.  Fat can be formed or stored excessively in the liver in a variety of conditions.  Excessive alcohol use puts fat in the liver.  Some patients with Hepatitis C store excessive amounts of fat in the liver, and when this happens, the Hepatitis C virus is harder to eradicate.  People with diabetes have metabolic errors which put excessive fat in the liver.  People who are obese may store fat excessively in the liver.

Another answer is gastroesophageal reflux.  If a person is obese and has a thick layer of abdominal fat, the weight of the abdominal wall can squeeze on the stomach, slow down the rate of stomach emptying, and increase the reflux of stomach contents up into the esophagus.  In other words, more food remains in the stomach, and the weight of the stomach pushes more material up into the esophagus.