Soda, the gas carbon dioxide, is forced into solution in the liquid. When the lid is stays closed, the gas stays in the solution, and the pressure stays high in the bottle. But when the cap is released, the carbon dioxide comes out of solution and makes the bubbles characteristic of soda. In the other words, carbon dioxide gas is mildly acid, which combines with the water in the soda to form carbonic acid. This acid, carried up and out of the bottle as the bubbles escape, can make your eyes burn. They then water as a defensive mechanism to keep the acid from damaging the surface of your eyes. It can also make your mouth and throat burn slightly as you drink the soda itself. If you want to know more about carbon dioxide, you can check https://www.angelfire.com/cantina/carbonation/
Basically speaking, drinking too much carbonated soda will make you fat because it will stimulate your appetite. However your situation is weired. Maybe your eyes nerves are irritated after drinking the carbonated soda which then indirectly influence your system of tear glands. You'd better go to see the doctor and have a check.