Miller, who earned 21 caps for Ireland and played for 12 clubs including Manchester United, Leeds and Sunderland, had been battling cancer.
Reports emerged in November that Miller had pancreatic cancer and was receiving treatment in the US, where he played for Wilmington Hammerheads.
Tributes for the Cork-born player have poured in on social media.
Celtic tweeted: "Everyone at #CelticFC is deeply saddened to hear of the passing of former midfielder Liam Miller. Our thoughts and prayers are with his family and friends at this time. RIP Liam, YNWA."
Former Ireland teammate Steven Reid said: "RIP Liam Miller. Was lucky enough to room with Liam on a few international trips.. top player and a great person. Thoughts and prayers with his loved ones."
Former Manchester United defender Danny Higginbotham said: "RIP Liam Miller. Devastating news. Great player but more importantly fantastic person. So sad. Thoughts with his family."
Hibernian FC said: "We are incredibly saddened to hear of the passing of former Hibernian midfielder Liam Miller. Our thoughts are with his family and friends at this time."
Miller made his professional debut for Celtic in 2000 after coming through the youth ranks at Parkhead and helped them win the Scottish Premier League title in 2004.
He then moved south to Manchester United on a free transfer, making 22 appearances under Sir Alex Ferguson in two seasons - which included a spell on loan at Leeds - before joining Sunderland in 2006.
Miller was part of the Black Cats' squad that won promotion to the Premier League in 2007 but he moved again in 2009 to QPR.
A brief spell at Loftus Road was followed by stints at Hibernian, Australian clubs Perth Glory, Brisbane Roar and Melbourne City, home-town club Cork City and, most recently, Wilmington.