A 1975 episode titled The Germans was taken off UKTV's streaming service because it contains "racial slurs".
In it, the Major uses highly offensive language, and Cleese's Basil Fawlty declares "don't mention the war".
Cleese wrote on Twitter: "The BBC is now run by a mixture of marketing people and petty bureaucrats."
He added: "I would have hoped that someone at the BBC would understand that there are two ways of making fun of human behaviour.
"One is to attack it directly. The other is to have someone who is patently a figure of fun, speak up on behalf of that behaviour."
He went on to compare the situation with that of Alf Garnett, the racist character in sitcoms Till Death Us Do Part and In Sickness and in Health.
"We laughed at Alf's reactionary views. Thus we discredited them, by laughing at him," Cleese wrote.
"Of course, there were people - very stupid people - who said 'Thank God someone is saying these things at last'. We laughed at these people too. Now they're taking decisions about BBC comedy."
He continued: "But it's not just stupidity. The BBC is now run by a mixture of marketing people and petty bureaucrats. It used to have a large sprinkling of people who'd actually made programmes. Not any more."
"So BBC decisions are made by persons whose main concern is not losing their jobs... That's why they're so cowardly and gutless and contemptible. I rest my case."
John Cleese responds