You should get this checked out. Burning pain is a sign that you're having nerve problems. Nerve pain in the feet is sometimes the first sign of diabetes - something you definitely want to catch early.
Burning sensations in the heel can be due to something called plantar fasciitis which refers to inflammation in the tendon that runs along the bottom of the foot. Those sensations are usually limited to the heel area, however.
I'm definitely not an expert but having burning pain in two different areas of the foot suggests the cause may not be anatomic in nature; ie not due to something on your nerves.
Be sure to get this checked out. Good luck!
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