The term "heart burn" is very misleading, as it has nothing to do with the heart. It is a stomach problem that occurs when stomach acid comes into contact with the oesophagus. The oesophagus is the tube that food goes down when you swallow and ends at the stomach.
Diabetics can sometimes suffer with heartburn as a result of damage to the nerves that control the lower oesophogeal sphincter, or the valve at the entrance of the stomach.
This means that after a meal, especially if lying down, the lower oesophageal sphincter does not tighten sufficiently and can let some acidic stomach content back into the oesophagus that can burn and feel irritated or painful.
A condition caused by the same degeneration of the vagus nerve is gastroparesis which is the inability of the stomach to empty, and push food into the intestine. If this occurs, the stomach will stay fuller for longer, opening up more opportunities for heart burn.