Ireland’s deputy premier Micheal Martin said Sir Tony had an “extraordinary impact on Irish business, sport, media and society”.
In a statement, his family said: “In the coming days there will be many worthy tributes made to Tony O’Reilly’s unique and extraordinary achievements in the fields of business and sport.
“As well as to his extraordinary philanthropic vision which was best evidenced by the establishment of the Ireland Funds at a dark time in this island’s history.
“But, for us, he was a dearly-loved dad and a granddad.
“He lived one of the great lives and we were fortunate to spend time with him in recent weeks as that great life drew to a close.”
Born in Dublin in 1936, Sir Tony made his international debut for Ireland in rugby in 1955 and soon became the youngest player to be selected for the Lions.
In his business career he pioneered the dairy brand Kerrygold, turning it into one of Ireland’s most well-known global consumer brands.
He later became the chairman of the food giant Heinz and in 1973 took control of Independent Newspapers, publisher of the Irish Independent, Sunday Independent and Evening Herald.
He was also known for his philanthropy, setting up The Ireland Funds, which gave money from US donors into reconciliation projects around the Irish border.
Sir Tony, who had joint Irish and British nationality, was knighted in the 2001 New Year Honours by the late Queen Elizabeth II “for long and distinguished service to Northern Ireland”.
Mr Martin said on X: “Saddened to learn of the passing of Tony O’Reilly a pioneering spirit who had an extraordinary impact on Irish business, sport, media and society.
“Through the Ireland Funds, Tony changed the global narrative on peace and reconciliation on this island.
“My deepest sympathies to his children, family and friends.”
As news of his death emerged, the Irish Rugby Football Union (IRFU) posted on X: “A legend of the game has passed. Our deepest sympathies to his family and friends.”
Sir Tony was the father of six children.
Mr Harris said ‘‘his passing will be felt by the many people he encountered in his long life My thoughts are especially with his children and grandchildren at this sad time.”
Mr Harris added: “In a life so varied it is hard to focus on one area, however Tony O’Reilly’s founding of the Ireland Funds with the late Dan Rooney had a transformative effect on the island of Ireland. The Ireland Funds is an example of peace and reconciliation done brilliantly from the ground up in communities.
“It has been one of the largest donors to our shared island and the cause of peace and is still going strong.’’