The popular Irishman started his career at Celtic in 2000 before moving to Manchester United four years later.
FAI chief executive John Delaney added: "It is very upsetting to learn of the passing of Liam Miller at such a young age".
"On behalf of all my staff who worked with Liam at the time, we send our deepest sympathy to his family".
A statement from Cork Schoolboy League said their thoughts and prayers were with Liam's family.
"Liam was so young and it is devastating for his family, friends and colleagues to have lost him so early in life".
Miller also won 21 caps for the Republic of Ireland at worldwide level.
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Celtic held a minute's silence in tribute to their former midfielder before the Scottish Cup game against Partick Thistle on Saturday.
Fine Gael councillor Des Cahill told Independent.ie: "As a fan I particularly remember him playing for Celtic and latterly for Cork City".
"And, of course, when the news filtered through last night it was terribly sad, and especially when you think of his wife and three children".
"I am very sad to hear that Liam has died", the former-Manchester United manager said. "Our thoughts are with his friends, former teammates and family at this time".
In an interview with Cork examiner in 2015, Miller said: "I genuinely look back and think "I played for Celtic, my boyhood club".
Beckham tweeted: "Our thoughts are with Liam's family". Taken far too soon. We all heard the news this morning.
James Forrest, who scored a hat-trick in a 3-2 win for the home team, admitted the news had stunned the Celtic players and expressed hope the result had been a fitting tribute to the former player.