For people relying on "low carb" diets like the famous Atkins diet, this is a tough choice. They shall indeed be forced to choose between weight loss and a healthy heart. It seems that you can either lose weight with such a diet, or remain heart healthy by saying no to it!
Research shows that "low carb-high protein" diets increase the risk of heart disease. What happens is that such diets greatly increase the amount of plaque in the blood vessals of the heart. This increases the risk of heart attack and stroke. Such diets also reduce the body's ability to recover from a heart attack because they prevent the formation of new blood vessals in the body.
These findings are the result of a study conducted on mice. Though the Atkins-fed mice did not gain weight (unlike the other mice who ate normal carb-protein mixed food), they had bad vascular health in comparison.
So this is another reason why your should get more active and rely on exercise in order to lose weight. Go for it, people!