No. Burping has no real lasting effect. However if you have esophageal reflux (acid coming up into the esophagus), this can be harmful and you should be seen by a gastroenterologist.
Yes. Belching is the venting of gas from your stomach. This gas generally gets into your stomach by swallowing it (usually unintentionally). When you belch, the gas is released into your esophagus and to your mouth. Sometimes stomach acid can follow the air which can cause burning and occasionally damage to your esophagus.