The club's first-team coach originally contracted skin cancer in 2006 and although he received treatment, the disease returned in March.
Former Scotland international Burns, who spent 15 years as a player at Parkhead and also managed the club for three seasons, had been undergoing treatment in both Glasgow and France in recent weeks.
"Tommy, a true Celtic legend and wonderful man will be sadly missed by us all.
"Clearly, our thoughts are very much with Tommy's wife Rosemary and his family at this extremely difficult time."
Burns joined Celtic as a teenager in 1973 and went on to play 352 league games for the club, scoring 52 goals, and winning eight Scotland caps.
In 1989, he moved to Kilmarnock and was given his first job in management there three years later.
He left to take the Celtic reins in 1994 but was sacked three years later.
A short spell as boss of Reading followed before Burns became Scotland number two in 2002 under Berti Vogts and later Walter Smith.
He returned to Celtic as first-team coach shortly before Martin O'Neill's arrival, a position he retained when Gordon Strachan became manager.
He is survived by his wife of 28 years, Rosemary, and four children.
Celtic chairman John Reid described Burns as a man of "true integrity and dignity" whose death would be mourned by everyone connected with the club.
The former Home Secretary said on the club's official website: "The passing of Tommy Burns is devastating news and my sincere and heartfelt sympathies go to Rosemary and Tommy's family.
"He was a true gentleman and someone who had the ability to connect and engage with anyone he met.
"As Celtic supporters, we recognised Tommy as one of our own and Tommy in turn was always delighted to spend time in the company of fellow fans.
"It was a privilege to know Tommy Burns. He was a man who gave so much to the club he loved over so many years.
"Tommy's passing will be mourned by the entire Celtic family. He will be hugely missed by us all, but he is someone we will never, ever forget."
Britain's sports minister Gerry Sutcliffe also paid tribute.
"I'm very sad to hear of Tommy Burns' death," said Sutcliffe. "Tommy was a fantastic player who rightly became a Celtic legend. His death is a great loss for football. My thoughts go out to his family at this time."
Dundee United manager Craig Levein sent condolences from the Tannadice club.
He said: "I knew Tommy was seriously ill, but it is still a shock to hear this.
"He was a decent man and a real football person and all our thoughts right now are with his family."