Why are my nipples burning/stinging?

I am sorry to hear that you are dealing with nipple discomfort. It is not 
uncommon to experience this type of pain in the nipples after a breast augmentation. The stinging and burning pain you are experiencing can occur as a result of nerves being stretched due to your newly increased breast size. It can take the nerves some time to adjust to these physical changes. I recommend that my patients gently touch the nipples a few times a day in an effort to gradually desensitize them. If you are still concerned about the progress of your recovery, I recommend setting up an appointment with your surgeon. He or she will best be able to advise you regarding your healing process. Best of luck.

At only 3 weeks postop, you are still very early in your recovery.  Changes in nipple sensation are very common.  Changes in nipple sensation can sometimes take the form of decreased sensation, and other times it can be hypersensitivity like you are describing.  It is normal to
have soreness and what are often described as "zingers," or sudden
nerve type pains.  This is often
accompanied by nipple sensitivity due to stretching of the nerves.  Fortunately, the symptoms you are describing
typically resolve in a few weeks or months. 
At 3 - 6 months postop, you will have a good idea of how your
breasts will look and feel long-term, but it can take a full year to
achieve complete healing sometimes.  I
would recommend following up with your plastic surgeon to discuss your
concerns.  Good communication and rapport between patient and surgeon is
very important both before and after surgery.
Ziyad Hammoudeh, MD