“Does constant coughing indicate a heart problem?”

Unlikely. The most common cause of night coughing is because of postnasal drip (dripping of mucous in the back of the throat from the sinuses). It typically indicates chronic fluid accumulation in the nasal sinus (you might need to be checked by an ENT physician). That being said, certainly patients with heart failure or weakness of the heart muscle who retain fluids can have nocturnal orthopnea (shortness of breath at night).