Joshua, 27, knocked out Wladimir Klitschko in May to add the WBA title to the IBF strap he already held.
But Tyson - a two-time world heavyweight champion - has warned the position can be "a crown of thorns".
In a wide-ranging interview with BBC Radio 5 live boxing's Mike Costello, Tyson outlined his belief in Joshua's ability, gave his views on a potential Floyd Mayweather-Conor McGregor bout and offered insight into the man who turned him into the youngest heavyweight champion in history.
Tyson describes Joshua as a "happy fighter" who is "fast for a big guy", and his admiration grew when the Briton survived the first knockdown of his career to stop Klitschko in the 11th round at Wembley Stadium.
He places huge emphasis on the heart Joshua showed to "finish the job", insisting such grit is the most important quality in the sport.
"Joshua has the potential to do a lot of things," said Tyson, 50. "He's got the look and throws a lot of hard punches. But there is so much pressure on him. I was in Dubai and there are big posters of him there."
Tyson was stopped from entering the UK in 2013 because of previous convictions, one of which saw him serve three years of a six-year jail sentence imposed in 1992 for raping a teenage beauty-pageant contestant.
In a career packed with incident, he also regained the WBC world title after his spell in jail, bit the ear of Evander Holyfield during a 1997 bout and filed for bankruptcy in 2002, three years before his final fight.
He told 5 live's boxing podcast: "The heavyweight championship will drive people crazy, you know that right? It's like a crown of thorns. Everyone wants to use you for something. It's like being the President of the United States.
"Joshua can't get the big head. He has to focus on fighting. When you start focusing on money, girls or whatever it is, it's going downhill. No religion, nothing, you can do those things when the fight is over. Let's see if he can handle that stuff.
"It's his time. I could be wrong, I'm not the gospel of boxing but he impressed me with Klitschko."
A similarity between Tyson and Joshua can be drawn as both men found boxing after difficult spells in their youth.
By his own admission, Tyson lived in the "hell hole" neighbourhood of Brownsville in Brooklyn, New York. Joshua meanwhile, pled guilty to possession of cannabis with the intent to supply just over a year before he won Olympic gold in 2012.
Tyson, whose father had left and mother had died by the time he was 16, found Cus D'Amato - who became his legal guardian, taught him to read and write, and coached him on the road to boxing stardom.
His new book, 'Iron Ambition', focuses on his relationship with D'Amato, who Tyson says forced fighters to visualise success and believe in themselves.
"I never had a father figure but I knew what it was like to have a father being with Cus," he said. "My objective was to make my father happy. He wanted to have a heavyweight champion of the world and that's what I wanted to do.
"That's the 25-million-dollar question. How did I get hooked to that old white Italian guy? What did he say to me to make me believe this other life is over and I stick with the boxing world? He made me worship the fighters.
"I came with no self-esteem and he turned me into a megalomaniac. I know he gave Muhammad Ali instruction for the George Foreman fight. He was an amazing man.
"He'd give you the reaffirmation that you are the greatest fighter in the world and no man can beat you."
D'Amato died a year before Tyson became the youngest heavyweight champion in history at 20, beating Trevor Berbick in two rounds he believes his tough late trainer would have called "sloppy".
Iron Mike had fought 28 times in 20 months as a pro to land the WBC title but concedes the loss of the man guiding him ushered in a "dark period".
"Everyone else was trying to get me for their own reasons, I was just lost without him," added Tyson, who took champagne to D'Amato's grave after his title win.
"I'd be a bum, imagine what I'd be without meeting that guy. I don't think I'd be in good shape. I'm so happy I met that old gentleman. He was a good man."
Almost 12 years have passed since Tyson ended his career with 50 wins and six losses.
Alongside television work he still studies the sport and talks up Tyson Fury's credentials when prompted, insisting the former world heavyweight champion "really knows how to fight".
His belief in the boxing craft of Conor McGregor is not so apparent when talk turns to the potential meeting of the two-weight UFC champion and Floyd Mayweather.
"Conor is a personality," added Tyson. "It's very rare to get a personality like that. Floyd doesn't have that.
"I don't believe that is going to happen. If it's boxing then Conor doesn't stand a chance. He would in UFC but if it's a boxing match it's not going to be a fight."
Asked if he would still attend, his response is energetic: "Yes, I would."
Clearly the man who admits he would watch bouts day and night as a youth is still hooked on the sport which plucked him and countless others from difficult beginnings.