It turns out that carbon dioxide and water together also undergo a chemical reaction to form a very weak (dilute) solution of carbonic acid.
CO2 + H2O ↔ H2CO3
Weak acids usually taste sour, like a lemon, but there is so much sugar in soft drinks, and so little acid from the carbonation, that it is sometimes difficult to tell.
After much consideration, my best educated guess is that when you burp, a little liquid comes up in that burst of air from your throat and that liquid has a little carbonic acid in it. The tissue in your sinuses is VERY sensitive, so the acid makes it feel like it is burning.
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