Lots of things happen to him ... some more obvious than others. In the long run, his guilt eats away at him until he dies. There is not direct link to this, just as there is no proof that anything appears on Dimmesdale's chest at the 3rd scaffold scene, but it is clearly suggested. His social guilt, guild in the eyes of the townspeople, keeps him from Hester, the one person who could have brought some sanity and peace to his life (suggested in the forest scene in Chapter 17).
The most interesting thing that it I think it does is make him a better minister. His understanding of sin, of human fallability, makes him so understanding of the weakenss and sins of others that his failing actually helps his ministry. This is not an uncommon theme in Hawthorne. In "The Minister's Black Veil," Hooper becomes a better minister because of his own failings. In "Young Goodman Brown," Brown is destroyed by the fact that he cannot understand that people are ambiguous, some good, some bad.
Guilt didn't make Dimmesdale's life better, but through it, he became a better Minister and made other folk's lives better.