Irene: As your profile suggests that your English is advanced, I'll presume that you know what 'breaking my heart' means, and I'll presume that you're asking about the construction with 'go'.
"Don't go + ...ing" is an informal, idiomatic construction used in the negative imperative form. We use it when we think that another person is likely to want ( or be tempted) to do something that we don't want them to do. For example, let's say that a small boy is going away on a weekend trip, and his mother gives him some pocket money. The mother might say "Don't go spending it all at once. That money's got to last you for the whole weekend'. She suspects that the boy is likely to spend all his money at once, and she knows that she can't prevent him doing this once he has left for the trip.
In the case of this song, "Don't go breaking my heart" is more of a plea. It shows the vulnerability of the singer. He or she has no control over what the other person might do in the future, but is begging them not to be hurtful.
"Don't Go Breaking My Heart" = don't emtotionally hurt me because I love you
And for your listening and learning English pleasure :)
"Don't Go Breaking My Heart" with lyrics!
Backstreet Boys - https://www.youtube.com/watch?v=r891C5EXWmI
Elton John and Kiki Dee - https://www.youtube.com/watch?v=1XcWRl8l9vE
