He studied at Burgos, and was a priest by 1627, being in that year an honours student at the Dominican College of St. Thomas in Rome.
From 1629 to 1647, Burke was simultaneously vicar apostolic of Kilmacduagh and rector of Holy Cross College at Mont Cesar, Louvain (1634–36).
He was in Galway in 1642, working as an intermediary between Patrick D'Arcy, Richard Martyn - representing the disaffected citizens of the town, and Clanricarde.
He departed from Galway in March 1651 as emissary to Innocent X, and spent much of the following ten years in France.
O Floinn writes that: "His cosmopolitan career ... vision, initiatives and principal role have yet to be fully recovered and historically reappraised."