It is possible that heartburn is the cause of your cough but burning can be a pretty non-specific symptom and we should not assume that that is what is going on. The most important thing is to work with your doctor, to attempt to find the cause of the cough. A new onset cough, especially in the absence of the signs and symptoms of a Cold, must be taken seriously by both you and your doctor(s). Most of the time, the cause of such a cough will be found to be benign and treatable but we can’t assume that either.
The first step should be a careful history taken by your doctor, followed by a physical examination and then a chest X-ray (frontal and lateral views.) If the X-ray is normal/negative, and if the history provides no clues as to the cause (for example post-nasal drip or asthma or recent exposure to allergens or irritants or infection) further examination of your respiratory tract, from sinuses down to the bronchial tubes in your lungs would be the next step. It would also be appropriate to determine if you are having acid reflux and, if so, consider a trial of therapy for that.
If your doctor is unable to establish a cause, with certainty, you and he/she should request consultation with a lung specialist.
Good luck