Biddy Byrne or Barns was my 4th g-grandmother, but she seems to have worked in the household of Johnston Moreton, a wealthy landowner in County Leitrim. At this time, I don't know where in County Leitrim. This would have been in the latter half of the 18th century.
In the early to mid-19th century, there was a Burns family in the Mohill area who married into another line of my tree. I don't know if these folks were linked to Biddy Byrne/Barns, and I don't know if these County Leitrim people fall under the umbrella of Ó Beirn.