A post on Mr Warr’s Twitter page announced his death from pancreatic and liver cancer on Saturday morning.
Mr Warr featured as the headmaster in the Channel 4 series That’ll Teach ‘Em, a programme which placed a group of 16-year-olds in a recreated 1950s state grammar school.
After being accused of sex offences in 2012, Mr Warr was found innocent and then campaigned against false allegations of sexual misconduct, writing a book on the subject called Presumed Guilty.
The post on his Twitter page said: “"It is with great sadness that I've been asked to let all Simon's followers know that he passed away early this morning at the age of 65 after a short battle with pancreatic/liver cancer.
"He was surrounded by his family, as well as the love & prayers of his many friends,” it added.
The message also said that Mr Warr had recently finished a new book.
Two days before his death, Mr Warr wrote on Twitter that he was "unlikely to be on Twitter again” and that he was receiving care in a hospice.
In his last post, he said: "I like to think that my legacy will be a greater awareness of the human costs of false sexual allegations & wrongful convictions.”
The BBC radio presenter Jeremy Vine paid tribute to Mr Warr, saying he had “incredible zest and a passion for learning and debate.”
BBC Radio Suffolk, for whom Mr Warr made regular appearances, said: “He was a ‘one-off’ with a huge sense of fun. We will miss him so much.”
An additional statement from the BBC said Mr Warr was "a much loved member of the team at BBC Radio Suffolk.
"We will miss him very much and our thoughts are with his family."