His agent Lynda Ronan said he died "peacefully" and left a "host of friends who were his chosen family".
Kelly acted in On The Up and Barbara, and recently played The Wizard in West End musical Wicked.
Ms Ronan said: "Sam Kelly died peacefully this morning after a long illness bravely fought. He does not leave any family but a host of friends who were his chosen family. His death is a great loss to them and the profession."
Stephen Fry led the way in an outpouring of grief on Twitter:
Very saddened to hear about Sam Kelly’s death. He played a splendid Hitler in a comedy drama called Stalagluft I made with Nick Lyndhurst…
— Stephen Fry (@stephenfry) June 14, 2014Cannot believe that lovely Sam Kelly has died. We were at drama school together - lovely, funny , talented chap. So sad.
— Colin Baker (@SawbonesHex) June 14, 2014Desperately sad news that the wonderful Sam Kelly has left us. Such a funny, talented man and one of the good guys. RIP Bunny Warren
— Mark Gatiss (@Markgatiss) June 14, 2014Sam Kelly gone. I was lucky enough to work with him. He was a gent and a darling. And he will be missed.
— amanda abbington (@CHIMPSINSOCKS) June 14, 2014Feel so depressed about the amazing Sam Kelly's passing how do we manage a second comic musical tonight?we dedicate it to him RIP
— Robert Lindsay (@RobertLindsay) June 14, 2014Such sad news about Sam Kelly. I got to know him slightly only recently. Such a sad loss.
— Michael Starke (@MichaelStarke57) June 14, 2014Totally devastated to hear of the sad news of my friend and 'Wizard' Sam Kelly's passing!!! A truly wonderful and incredible gentleman! x
— Natalie Anderson (@AndersonNatalie) June 14, 2014Sam Kelly. What a lovely actor.
— Rebecca Front (@RebeccaFront) June 14, 2014Sam Kelly. The world is a poorer place without you. And what a talented versatile actor. RIP.
— Graham Seed (@Grahamseed) June 14, 2014