The answer is: Not very often.
Horses do not generally get coronary artery disease ("heart disease"). Herbivores are simply not prone to this condition, even if overweight. Overweight horses get other heart problems, but they don't get heart disease.
Therefore, they don't get heart attacks (myocardial infarction). It simply doesn't happen.
However, the term heart attack is also commonly used to refer to animals (and people) who die from Swale syndrome or sudden death syndrome.
Swale syndrome is the defect that often goes undetected until it strikes down young, apparently healthy human athletes. It occurs in horses as well. Just as in humans it's seen in the athletic, in horses it's seen in actively competing sport horses and race horses. The famous show jumper Hickstead who collapsed and died in the middle of a round at a major show was one recent example.
There's no way to predict the disease (Feel free to use it as a plot device) and it's generally not possible to save the animal.
But if you're looking for ways to kill off old horses, heart disease isn't it. More on that tomorrow.