In 1991, he was assigned to parish ministry, serving also as vice-chancellor and chancellor of Grand Falls diocese. He became a faculty member at St. Peter's Seminary in London, Ontario, in 2002 and dean of studies three years later.
Pope Benedict XVI named Dunn auxiliary bishop of the Diocese of Sault Sainte Marie in Ontario in July 2008.
Benedict XVI named him the bishop of Antigonish, Nova Scotia, a year later, replacing Raymond Lahey, who was charged with the importation of child pornography in 2009 and dismissed from the clerical state in 2012. Dunn was installed as bishop of Antigonish on Jan. 25 2010.
Dunn took part in the 2012 Synod of Bishops on the New Evangelization in Rome.
He spoke at the synod of the need to evangelize victims of clerical abuse. He also called for "a deliberate and systematic involvement and leadership of women at all levels of Church life, e.g., permitting women to be instituted as lectors and acolytes and the institution of the ministry of catechist."
On April 13, 2019, Pope Francis appointed Dunn as the coadjutor archbishop of the Archdiocese of Halifax-Yarmouth. He continued to serve as apostolic administrator of Antigonish diocese until a new bishop was appointed in Dec. 2019.