I found this information somewhere and will pass it on.Since it references published literature I have no reason to believe it's false:
"The Dictionary of Irish Family names by Ida Grehan: MacBirney with variations MacBiorna, Birney, (Mac)Burney, and MacBirnie. MacBirney and its many variants evolved from the Norse personal name Bjarni-the Norwegian surname for bear is bjorn. It is said to be the Irish equivalent of MacMahon, or Mac Mathghamhan, which means son of a bear. It is essentially a Scottish name. In Dumfriesshire it is usually shortened to Burnie, while in other parts of that country it is to be found asMacBirnie, MacBurnie, and even Macfurney.
It was not until the 17th century that it first appeared in the Acts of Parliament in Scotland, although a David M'Birney is mentioned as a witness in Kirkcudbright in 1466, while a Cuthbert McBurnie was appointed the "Brigmaistre" of Dumfries in 1520-an area where there is a concentration of the surname.They must have come as mercenary soldiers or planters to Ulster, where McBurney is still a common name. It is rarely found in other parts of Ireland.
Similar information is given in the book Surnames of Scotland by Black with a few other historical citations."