Tinker, 58, was taken ill on Sunday at his Brighton home and died in the Royal Sussex Hospital. It is believed he had a heart attack.
As theatre critic of the Daily Mail since 1972, Tinker's prominent overnight reviews, carried at the top of an early news page, were among the most widely read of any theatre reviewer in Britain.
Sometimes sporting a pony tail and always ready with a flurry of theatrical anecdotes, he was as well known to theatrical producers, actors and the casts he was reviewing.
Despite his outward bonhomie, Tinker's life was affected by tragedy. His youngest daughter Charlotte, an epileptic, drowned in the bath six years ago at the age of 24. Among those who consoled him was his close friend, the late astrologer Patric Walker, who took Tinker to stay with him in Greece.
Last night the biggest names in theatre spoke of their affection for Tinker and their shock at his loss. The West End producer, Sir Cameron Mackintosh, who had known him for around 30 years, said: "He was probably one of my longest and closest friends in the theatre. I can't believe the news. He was a friend, confidant and critic, a curious mix."
The composer Sir Andrew Lloyd Webber said: "Jack was one of the few people who remained a true friend while being a totally impartial critic."
His newspaper said in a statement: "Jack, an irrepressible character whose enormous talent earned him a string of awards, had just returned from a visit to America. He had a long and distinguished career covering the world of showbusiness and theatre and reviewed all the major show and play openings for the Mail."
Daily Mail deputy managing editor, Garth Burden, added: "He was a wonderful journalist and will be sadly missed here.
"He was a very approachable person and did a wonderful job for the newspaper. Everyone here is a bit shocked by it all."
Obituary, page 12