His comment has been seen as a swipe at left-wing party leadership rival Jeremy Corbyn, who is 66 and has been an MP for 32 years.
Mr Burnham, 45, said: “I think modern politics is intense – it’s changed in my 14 years in Parliament. I always felt I would give it my all for 20, 25 years. If you’ve had a seat for 25 years, people should let some new thinking in.”
A source close to Mr Corbyn described the shadow health secretary’s comments as “unhelpful” and claimed that they showed a “hint of ageism”.
However, a spokesman for Mr Burnham insisted the comments were not intended as an attack on Mr Corbyn.
The row yesterday came amid a series of personal insults levelled at Mr Corbyn, who entered the contest as a 100/1 outsider to “widen the debate” but is now expected to at least top the first round and possibly win overall.
Former Labour prime minister Tony Blair last month said Mr Corbyn’s supporters need a “heart transplant” and warned that he risks taking the party back to the 1980s.
Mr Burnham also launched a direct attack on Mr Corbyn’s economic policies yesterday.
He said the contest over the Labour leadership was now “about the big ideas”, warning party members: “I do not believe we lost the election because we were not left wing enough.”
Mr Corbyn hit back by claiming the country was still “paying the price” for the Thatcher administration’s policies.
He said: “I’m accused of being a throwback to the 1980s, but I’ll go back one decade further and just say there are lessons to be learned from what that government was trying to do.
“To recognise the changing industrial face of the whole world also recognises the huge skill levels we have and the traditions of industry we have in this country that were allowed to be wasted on the altar of monetarism and the development of a financial services economy, rather than a manufacturing economy.”
Dismissing the personal criticism aimed at him, and rumours in Westminster that fellow Labour MPs could seek to oust him if he won the poll, Mr Corbyn said: “It is the name-calling, the depoliticisation of serious political debate that drives people away.
“Our campaign is not getting involved. We aren’t doing rebuttals. We’re not interested.”
Meanwhile, fellow leadership hopeful Yvette Cooper enjoyed a boost to her campaign after former home secretary Alan Johnson endorsed her.
He said: “I believe that Cooper has the intellect, the experience and the inner steel to succeed in this most difficult of roles. In my view only Cooper can unite the party to win again.”
In a sign of Labour’s difficulties, community organiser Arnie Graf, who was brought over from the US to help under Ed Miliband’s leadership, revealed that, for one event, local activists had not been able to find a worker on the minimum wage to appear with the then leader.
In an article on the LabourList website, Mr Graf said he was able to locate someone to take part in the event but added: “How could it be that the Labour Party, supposedly the party of working people, was not in relationship with a single minimum wage worker?”