The pre-election row broke out after Mr Cameron used Commons Question Time to highlight the Deputy Labour leader's remarks from a radio interview on Monday.
In it, Ms Harman spoke about public services used by lower- and middle-income families. During a response, she said: "Yes, people on middle incomes should contribute more through their taxes. But, actually, they need those public services like the transport system."
The PM told MPs: "That is their policy; the squeezed middle will be squeezed more". Referring to Labour leader Ed Miliband, he said: "Now he needs to tell us which people are going to pay which taxes ... Labour will put a tax on your job, on your mortgage, on your home and on your pension. So tell us where are the middle-income taxes coming?"
The Labour leader dismissed Mr Cameron's claims as "totally desperate stuff" because he had nothing to say about the cost-of-living crisis.
Later, a senior aide to Mr Miliband denounced Mr Cameron's comments as "deeply dishonest" and a "lie", accusing him of deliberately misrepresenting Ms Harman's words.
"Harriet Harman was clearly talking about the tax system as it is now, where people on lower incomes pay less tax," he said, adding: "It is the Tories who have raised tax 24 times - including raising VAT for hard-working families despite promising not to - and cut them for millionaires."
Shortly afterwards, a clearly angry Mr Miliband said: "Harriet Harman was making absolutely clear that we are in favour of fair taxes and we want to cut taxes for lower- and middle-income people and that's why we want a 10p starting rate of tax. This is pretty desperate stuff from the Tories."
At the same event, Ms Harman accused the PM of a "pathetic and deliberate misrepresentation" and of "low politics". She called on him to "apologise for misleading the House of Commons".
The deputy Labour leader insisted what she was referring to when she said middle-income people should pay more tax was the current system and that she was not advocating raising taxes higher. "That was absolutely evident, whatever the level of IQ in Tory Central Office ... the British people deserve better in a General Election campaign in which the Tories try and get votes by completely deliberately distorting and misrepresenting what Labour has said."
Asked if she was angry, Ms Harman replied: "I'm disappointed. The Prime Minister should be better than that."
But an hour later, Mr Cameron tweeted a new election slogan, using a picture of Ms Harman with her interview quote and the words: "Labour finally admit it: they want to put up your taxes."
It then emerged the deputy Labour leader had written a letter to the PM, saying his assertion based on her remarks was "not true; it is a lie" and that all she was doing was defending Britain's progressive tax system.
But Grant Shapps, the Tory Chairman, brushed aside Ms Harman's protestations, saying: "It's clear Ed Miliband would crush Middle Britain. He wants a new tax on your family home, higher fuel duty, a rise in corporation tax, and an extra tax on your pension. Nothing has changed. Labour are still economic vandals who would put your job at risk."
l Ed Miliband is expected next week to visit the White House for the briefest of meetings - a so-called "brush-by" - with President Barack Obama.
The Labour leader, who will be in Washington for a speech, has reportedly been lobbying hard for a meeting with the US leader, which it is thought would help him present a more prime ministerial image on the world stage ahead of the General Election.
Former Obama advisor David Axelrod is Labour's election strategist, but Labour described the possibility of an Obama meeting as just "speculation".