No, the heart hurts yes. Proof of this is the intense and prolonged chest pain experienced during a heart attack ("heart attack"). This heart pain is caused by lack of oxygenation of the heart muscle, a condition called ischemia.
When the arteries that irrigate the heart are blocked, the blood (and with it oxygen) does not arrive as it should to the heart muscle, which is then forced to produce energy through anaerobic reactions (without the presence of oxygen).
This leads to the production of lactic acid, a substance capable of stimulating the sensitive endings of the heart muscle, causing a sensation of burning discomfort or pain.
In addition to the infarction, there are other heart diseases that may also manifest through chest pain, such as inflammation of the pericardium (membrane covering the heart), heart valve problems and diseases of the aorta.
It is true that the heart does not possess all kinds of sensory endings than the skin, for example, and therefore can not produce the same types of pain. However, under certain circumstances the heart hurts and the pain can indicate a serious illness.
Thus, any and all type of chest pain should be reported to a general practitioner or cardiologist.