From your description, it must caused by nervousness. Just relax and keep inner peace, you can get rid of nervousness. Also, the symptom watery eyes can gone with your nervousness. Also, you control your eye blink so as to reduce water in the eyes. You shall prative more what you going to do so as to increase confidence, thus to keep calm. Good luck.
It sounds like that your body is reacting to your nervousness. You shall practice speaking more with eye contact to your friends to help you reduce nervousness. Also, as you remind of burnning in the eyes, it sounds like eye infection. Or have you spend too much time in front of computer screen? If so, it is normal to get watery and burnning eyes. And it is a accident that these symptoms happen when you talk to people. Don't worry, Take a deep breath and a good rest.