I heard drinking soda and belching could permanently damage your nose by burning it. Is this true?

No. Although this might make you feel uncomfortable, or embarrassed if witnessed by your friends, you cannot permanently damage your nose or sinuses this way. This is an urban legend. So eat all the pop rocks you want while drinking pepsi- that's an urban legend too.
Soda and nose. Belching after drinking soda is due to release of carbon dioxide (from the soda) and swallowed air accumulated in the stomach. The release of gases during belching, especially forcefully, can lead to a burning sensation in the nose. Forceful belching may also inadvertently aerosolize stomach acids and add to the burning sensation. There is no evidence to confirm permanent damage to nasal mucosa.