Miller, who won 21 caps for the Republic of Ireland between 2004 and 2009, had been suffering from cancer.
Miller started his career at Celtic in 2000 before moving to Old Trafford four years later, also playing for Leeds, Sunderland, QPR, and Hibernian.
Celtic said they were "deeply saddened", adding that their "thoughts are with his family and friends".
BBC Scotland pundit Tam McManus, who played with Miller at Hibs, tweeted:
I’m afraid it is true that Liam Miller has sadly passed away today. Thoughts are with all his family and friends at this horrendous time. So sad.— Tam McManus (@The_Tman10) February 9, 2018
Miller ended his career in the United States in 2016, having also played for three clubs in Australia and his hometown club Cork City before retiring.
'A gentleman, a friend, a great player'
Think you can see the esteem that Liam Miller was held in by supporters of every club he either played for or against with the tributes to him tonight. Such a wonderfully talented player but more importantly a lovely, humble person who never changed despite the wealth and success— Tam McManus (@The_Tman10) February 9, 2018
Kevin Kilbane, who was Miller's team-mate with the Republic of Ireland, tweeted:
Sad hearing the awful news that Liam Miller has passed away this evening, taken far so soon by a terrible disease. Proud to have shared a dressing room with him over the years. #RIPLiamMiller— Kevin Kilbane (@kdkilbane77) February 9, 2018
Liam was a great team mate & room mate. He was also a top opponent. But most of all he is one of the few people in football who I consider a true friend. I'm absolutely devastated for him and his young family. I'm sure he is resting with the angels now. RIP Chewy love you x 😭 pic.twitter.com/YB8b9UzJwt— Stephen Elliott (@SleevesElliott) February 9, 2018