He will take over on September 1, in time for the launch of the 10th season of tournament organised by Celtic Rugby two days later.
He will replace John Hussey, of the IRFU, who has acted as Celtic Rugby chairman for the past seven years.
Outgoing chairman Hussey said: "As the Magners League prepares to expand through the introduction of two Italian teams to join the 10 clubs from Ireland, Scotland and Wales, I am delighted the Celtic Rugby Board has recognised there is a growing need to support the tournament's expansion.
"Last season saw the highly successful introduction of a play-off system and grand final to determine our champions and the changes agreed by the Board are aimed at continuing to shape the ever growing reputation of the tournament.
"Having a man of the calibre of Andy Irvine to take over from me is a real statement of intent from the Celtic Rugby Board and all the changes we have agreed will help support the growth of the Magners League product and our 12 clubs."
In addition to the appointment of Andy Irvine as a non-voting Chairman at the head of an eight-man Board of Directors, John Feehan has been appointed Chief Executive of Celtic Rugby following the re-structuring of the Magners League organisation.
This means the tournament will now be managed out of the same Dublin office in charge of both the RBS 6 Nations Championship and the British & Irish Lions. These major and bold initiatives were all approved at a Celtic Rugby Board Meeting held in Cardiff in the day.
Feehan, who is currently Chief Executive of both the Six Nations and British & Irish Lions, will head up a team that will include 10 permanent members of staff, including the longstanding tournament director, David Jordan.
Celtic Rugby was born in 2001 when the then Celtic League was launched. There were nine Welsh, four Irish and two Scottish teams involved in the inaugural campaign and, as Celtic Rugby prepares to enter its 10th anniversary season, the Magners League will grow to 12 teams with the inclusion for the first time of two Italian franchises, Aironi Rugby and Benetton Treviso.
Last season the inaugural Magners League grand final at the RDS between the Ospreys and Leinster was a sell-out occasion that featured 14 British & Irish Lions.
The vast improvement in the standard of both domestic and international rugby in the three Celtic countries since the Magners League was formed has led to Wales winning two Grand Slams, Ireland one and four major European titles, including three Heineken Cups, being won by teams from the tournament.
Now, in an effort to allow the competition to further grow and develop, the Board has put in place this new structure with the competition being run from the Irish capital.
The structure of the new organisation will include a Board of eight voting members (two from each participating Union) plus the CEO, tournament director and the finance director in a non-voting capacity.
The Board will also have a non-voting chairman to ensure that overall the well-being of the competition remains the primary focus.
"I consider that this latest episode in the Magners League's history will reaffirm the importance of this competition in European rugby. I wish John and his team the very best of luck in further developing what I believe is already a dynamic and engaging tournament," added Hussey.
New Celtic Rugby Chief Executive John Feehan declared: "It is my desire to do everything I can to see the Magners League competition reach its fullest potential.
"Not only do I believe that the competition can be pushed further, I think that my team and I are uniquely qualified and ideally situated to make that a reality.
"The taking on of the Magners League will undoubtedly pose a few challenges, but I know I speak for all of us in saying that we are ready to face them and are looking forward to getting involved with the game at club level."
The 2010 /11 Magners League campaign kicks-off on Friday, September 3 with matches at Ravenhill between Ulster and reigning champions the Ospreys and Firhill Arena, where Glasgow Warriors will host Leinster.