In a scathing intervention, the ex-Tory leader said the public were being led to expect a future outside the European Union that is “unreal and over-optimistic”.
Sir John called on the PM to inject “a little more charm and a lot less cheap rhetoric” into the Brexit negotiations.
Conservative MP Jacob Rees-Mogg accused the former premier of being a “bitter man”.
He told the BBC: “It was a craven and defeated speech of a bitter man who was heavily defeated by the electorate for his own failings in Europe in 1997, was defeated again last June and now wishes to take out his failures on Mrs May.”
Former party leader Iain Duncan Smith, a thorn in Sir John’s side over the EU during his premiership, said the speech was “peculiar” and “sad”.
He told BBC Two’s Newsnight: “What I thought when I looked at this speech was that this was a peculiar speech in the sense that it looked backwards the whole time.
“It was almost like a re-fight of the referendum, all the same threats and issues that came up during Project Fear were all in here.
“Strangely bitter, really, and almost, really, the speech of someone who simply refuses to accept that the British people should have made a decision such as they did, and wants them almost to rerun it again until they get it right, which is rather sad, really.”
In a speech at Chatham House in London on Monday night, Sir John said voters must be told how the outcomes of the negotiations will affect them.
“I have watched with growing concern as the British people have been led to expect a future that seems to be unreal and over-optimistic,” he said.
“Obstacles are brushed aside as of no consequence, whilst opportunities are inflated beyond any reasonable expectation of delivery.”
Sir John said the talks require “statesmanship of a high order” and warned of a “real risk” of the exit deal falling “well below the hopes and expectations”.
“I see little chance we will be able to match the advantages of the single market,” he said.
“In my own experience, the most successful results are obtained when talks are conducted with goodwill: it is much easier to reach agreement with a friend than a quarrelsome neighbour.
“Behind the diplomatic civilities, the atmosphere is already sour. A little more charm, and a lot less cheap rhetoric, would do much to protect the UK’s interests.”
The former premier’s time in Number 10 was marked by a series of bruising battles with his own MPs over Europe and his outspoken opposition to quitting the bloc has previously angered Eurosceptic Tories.
Sir John said Leave voters backed Brexit in the belief it would improve their lives.
“If events go badly, their expectations will not be met, and whole communities will be worse off,” he said.
“The particular fear I have is that those most likely to be hurt will be those least able to protect themselves.”
Sir John said it was “time to stop” the attacks on Remain supporters, insisting they are demeaning and intolerant.
A No 10 source said: “The Government is determined to make a success of our departure from the European Union and to move beyond the language of Leave and Remain to unite our country.
“The Prime Minister set out her 12 negotiating objectives for Brexit in January. We have a clear plan to get the best deal for the United Kingdom and are going to get on with the job of delivering it.”