During a heart attack, tissue in your heart muscle dies due to lack of blood flow through your hearts arteries. This is how doctors are allowed to determine if you have had a heart attack in the past.
The severity and number of heart attacks will damage your heart irrepairably. As much as your heart can survive a heart attack, the damage tissue will remain.

http://www.mayoclinic.com/health/medical/IM00938

(A). The plaques build over time (B) and eventually completely block your arteries and rupture (C), causing a heart attack. Damage tissue dies and your heart remains scarred(D).