Mr Burnham, former shadow home secretary, said the cost of bus travel in Greater Manchester means it makes more financial sense to order a taxi through the ride-hailing service than to take public transport as he warned of the potential for "complete gridlock on our roads" unless action is taken.
Labour is calling on ministers to consider introducing a concessionary reduced fare system for young people aged 16 to 19.
The party wants to amend the Bus Services Bill to require the Government to bring forward a national strategy setting out its vision for local bus services.
Shadow transport minister Daniel Zeichner said during the Bill's report stage that bringing forward such a strategy would show bus travel is as important to ministers as rail.
He said such a strategy should also look to "reduce the financial burden" on young people.
He said: "But we also believe the Government needs to do far more to support young people to help them afford the cost of bus travel.
"That's why we are asking that the Government include as part of a national bus strategy a consideration of a young persons concessionary fares scheme.
"Young people now have to stay in school, further education or training until they are 18, rightly so, and many of them use the bus to get there and we think it's quite right that the Government look at how they can reduce the financial burden on those young people who are only trying to get to get to their school, job or apprenticeship.
"While some local authorities still provide concessionary fares for young people, many do not."
Mr Burnham, who is now a backbench MP and running to be the mayor of Greater Manchester, intervened to highlight the prohibitive cost of bus travel.
He said: "Can I tell you on the issue of bus fares for young people, young people in Greater Manchester have told me that it is cheaper for them, for four of them, to get an Uber than it is to travel on buses in Greater Manchester sometimes, given the cost of travel.
"How on earth can that possibly make sense and how on earth can that lead to anything other than complete gridlock on our roads?"
The Bus Services Bill will extend the ability of local authorities to introduce franchising and make it easier to introduce multi-operator ticketing.
Critics have opposed the Bill on the basis that local authorities will not be able to form new municipal bus companies which would effectively allow them to run bus services in-house.
Conservative MP Huw Merriman said the idea should be looked at.
"Perhaps one consideration could be the overall cost of concessionary travel, and whether it's time for concessionary travel, perhaps for the over 65s, to be given purely to those who really can't afford it, so we're looking more at means testing," said the Bexhill and Battle MP.
Local government minister Andrew Jones dismissed Labour's amendment.
He said the Government recognised the cost of public transport could be an issue for young people, which is why a new bursary had been introduced to help disadvantaged students.
Mr Jones added: "The statutory responsibility for transport to education and training for 16 to 19-year-olds rests with local authorities.
"This enables them to make decisions that best match local needs and circumstances, with many authorities and operators already offering discounts for passengers in this age group."
Labour's new clause one, which also called for the Government to publish a national strategy for buses, was defeated by 278 votes to 193, majority 85.