I second the recommendation of keeping the appointment with the lung specialist.
Lungs don't hurt. The chest wall can hurt, and other structures (like your heart) can hurt. My guess is that your doctor thinks that your asthma isn't well controlled and that caused overuse of your chest muscles. If he thought it was your heart he would be sending you to a cardiologist.
It is prudent to follow your doctor's instructions for now, not least because he may know something that we don't know about your condition. On the other hand, the goal of treating asthma should be that with proper management your asthma doesn't limit you.
Your doctors should be helping you get to the point where you can run with their blessing.
Robin