What is heart pain that feels like heartburn?

GERD. Heartburn is not heart pain at all - the term is a misnomer. It is due to acid reflux in the esophagus and is called "gastroesophageal reflux disease". On occasion, actual angina (true pain arising from inadequate circulation in the heart) is mistaken for heartburn because some people can't tell the difference and/or the two conditions can feel similar. A stress test can separate them.
Depends. Burning chest pain can be related to gastrointestinal problems or to problems with the circulation to the heart. The term "heartburn" implies actually that it is gastrointestinal. But it is easy to confuse the two even for the doctor. So if you have burning chest pain, you need to get appropriate tests as the heart involvement is potentially very serious.