He added: "This is not an issue for scare stories and negative campaigning. It involves vulnerable people and we need to remember that at all times in this debate."
Mr Burnham has said he will be holding a conference this week with local authorities and charities to try to reach a consensus on how to pay for elderly care, to which the main political parties would be invited.
Lib Dem health spokesman Norman Lamb said he wanted to attend the conference and he accused Mr Lansley, who is refusing to go, of agreeing during their earlier talks that a compulsory levy was an option that had to be considered.
"Andrew's not being straight with people on this, because it was specifically recognised that there were both compulsory and voluntary options, and Andrew included that in his own draft statement of principles, and that's one of the key issues that we have to resolve," Mr Lamb told The Politics Show.
"Then it suddenly becomes the issue that results in the Tories abandoning the process and going for a very aggressive form of advertising."
Mr Lansley said the statement he had drawn up with Mr Lamb and Mr Burnham after their meetings had made it clear there were "differences" between them.
He later released a copy of the statement, which although it does not specifically refer to "death taxes" says: "We accept that within these shared principles, there is legitimate scope for differing views over the criteria for access to care and the respective contributions to meeting care costs".
He called on Mr Lamb and Mr Burnham to release the memos they wrote during the talks.
He also insisted there had been no agreement to suspend party political campaigning during their talks.
But he said it was Labour and not the Conservatives who broke the cross-party consensus, pointing to a leaflet he said had been sent out last week by Labour's Bolton North East MP David Crausby, which accused the Tories of scrapping the government's care plans.
But Mr Lansley admitted the "RIP" poster about Labour's plans had been based on a story in the Guardian newspaper, which had been denied by the government.
He denied it had been a pre-planned assault on the government, saying the poster had been "turned round" by Tory HQ within 24 hours of the story appearing in the newspaper.
'Lot to learn'
He also accused the government of using pollsters to test the idea of a 10% tax on estates to fund elderly care, which he said proved the "death tax" had not been ruled out.
Shadow Education Secretary Michael Gove has also weighed in with a personal attack on Andy Burnham.
The Conservatives' poster was based on a Guardian story
Mr Gove, who is two years older than the health secretary and has been an MP for less time, told BBC One's Andrew Marr programme Mr Burnham was a "young politician, idealistic, but he has a lot to learn."
The Tories have proposed a voluntary levy instead, which would guarantee free personal care.
Eighteen charities including Carers UK, the National Care Forum, Age Concern, Help the Aged, Alzheimer's Society and Macmillan Cancer Support urged the political parties not to reduce the issue to "election soundbites" and "poster slogans".
In a letter to The Times, they said: "The vexed question of who pays is unquestionably difficult, and the solutions may be controversial - but the costs of failing to act are simply too great to allow the debate needed to be drowned out by party-political squabbling."
Broadcaster and writer Dame Joan Bakewell, the government's ageism tsar, described the "political circus" as "shameful".
"It's highly regrettable that political interests have stepped in where a really serious issue was being discussed by serious men with the interests of older people at heart," she said.
The BBC's political correspondent Reeta Chakrabarti said "an attempt to take the politics out of personal care" had resulted in "bitterness and squabbling" and become a "huge unresolved problem".
"Perhaps it is no surprise, so close to an general election," she added.
Prime Minister Gordon Brown announced plans for free personal care at home for the most needy elderly people in England at the Labour Party conference last year, mirroring the system in Scotland.
The service - for people needing help eating, washing and dressing - is currently means-tested and there has been criticism that it penalises people on modest incomes who have saved for old age.
It is estimated that 280,000 people could qualify.
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