Dimmesdale never left Boston. He actually died at the scaffold of a combination of heartache, guilt, the strain placed upon him through physical stress, and the psychological strain of his life.
Dimmesdale was an ill man, weak both mentally, spiritually, and physically. He was a broken man because he failed himself, God, and his promise to his congregation. He was so guilty that he punished himself. He was such a broken man, that he knew his only way was to suffer somehow what Hester was going through.
What Dimmesdale did do was planto leave for Europe with Hester to escape the settlement of Boston, the punishments of Chillingworth, and the problematic situation in which Dimmesdale and Hester got into. Once the decision was made a big weight was lifted off Arthur Dimmesdale. He felt happy, joyful, hopeful. He was still ill from the years of past self-punishment, but both he and Hester felt that this would be a great solution.