The following report was published on the BBC News website:
Elliott started the magazine in London in 1968 and it grew into a global media brand covering hundreds of countries.
A statement on Time Out's website described him as "a visionary publisher, a tireless champion of city culture and a staunch friend".
It said Time Out's first post-lockdown print magazine in London on 11 August would be dedicated to him.
"It is with great sadness that we announce that Time Out's founder Tony Elliott passed away on 16th July, after a long illness," the statement said.
"He will be sorely missed by his family, friends and colleagues.
"His life and his work inspired millions of people who did not have the good fortune to know him personally."
Paying tribute to Elliott, BBC arts editor Will Gompertz described him as a "visionary".
"He really was the most wonderful, generous person whose passionate support for the arts was unstinting and invaluable," he wrote on Twitter.
Time Out Group's chief executive Julio Bruno said Elliott had died with lung cancer.
"He would not allow that to stop him," he wrote in an article on LinkedIn.
"He kept looking at the world with those inquisitive eyes, with that innate curiosity that very few possess in such measure."
Bruno said he met Elliott five years ago and he "was engaged with the company until the end".
"I will miss his advice, his passion, his profound understanding of the media world. And I will miss his friendship above all," he wrote.
"Tony was a visionary, a pioneer, a brave man and a great friend. We owe him very much and we will fight to keep his legacy alive."
David Fear, who previously worked as film editor at Time Out NY, tweeted that Elliott would "argue [with] half of our suggestions, smile and go 'Keep it up!'".
The magazine's global deputy film editor Joshua Rothkopf described him as "so Swinging London", adding: "I'll miss him."
Elliott founded Time Out while he was a student at Keele University.
The first issue of the listings magazine was produced from his mother's kitchen table, funded by £70 he had received from his aunt for his 21st birthday.
The company has now become a global media and leisure business covering food, drink, culture, travel and entertainment in 328 cities across 58 countries, through websites, magazines and live events.
Marking Time Out's 50th anniversary, Elliott said he started the brand "because it was hard to find out where to go and decide what to do in London: there was not one single place to find the information".
"So I effectively created a publication for myself."
In 2017 he was appointed a CBE for services to publishing. He was honoured at the British Media Awards with an outstanding contribution award that same year.