The former mayor of Greater Manchester has promised to stick to Labour's 2024 manifesto pledges not to raise income tax, VAT and national insurance but has suggested there could be "some movement" with possible increases to business rates on warehouses to fund tax cuts for pubs and some high-street businesses.
And when he launched his campaign to return to Parliament in May, Burnham indicated he wanted to change how property and land is taxed.
One of the measures Burnham has advocated in the past is scrapping stamp duty and replacing it with a land value tax (LVT) - an annual levy based solely on the value of the land itself, excluding any buildings or improvements on the property.
Another proposal he has reportedly backed is a proportional property tax - an annual tax based on a percentage of the property's value.
It's not clear that just because he's endorsed property tax reform that he'll definitely implement changes in government, but we've asked some experts for their view.
Stamp duty is a tax paid by home buyers on properties or land worth over £125,000, or £300,000 for first time buyers, in England and Northern Ireland.
The Institute for Fiscal Studies, an independent think tank, has described it as "one of the most economically damaging taxes", while Kemi Badenoch said last year a Conservative government would abolish the levy on main homes.
Burnham too has been critical of the tax, instead favouring an LVT.
"The LVT, an annual tax on the market rental value of the land, would allow for the abolition of stamp duty - a tax on the aspirations of young people to put down roots and get on in life," he wrote in the Guardian during his first bid for the Labour leadership in 2010.
In the 16 years which have followed, Burnham has repeatedly called for reforms to property taxes.
But being in favour of property tax reform and then enacting it in government when confronted with the political and practical reality are two different things.
Economists are broadly united in their contempt for stamp duty, largely because they say it penalises transactions.
"More or less any property tax or local tax is going to be better than stamp duty," says Stuart Adam from the IFS.
"What stamp duty does is discourage people from buying and selling."
He argues that a land tax is an efficient way of raising revenue because land is in fixed supply, visible and identifiable, and cannot leave the country.
There are no "damaging disincentive effects", with the LVT not having an impact on behaviour at all, he says.
He adds that one of the ways to deal with the shortage of the housing stock in the country is to make the best use of what we have - and scrapping stamp duty would help with that.
"It would lead to less pressure on the housing market - less need to build more stuff," he says.
But Lucian Cook, head of residential research at estate agents Savills, says implementing an alternative tax system would be fraught with difficulties - both practical and political.
"Has Burnham been given the electoral mandate to do it?" he asks.
"It is such a big change to the tax system with clear winners and losers; it will be difficult to pull off within the rest of this Parliament."
Cook argues that stamp duty is a simple tax because it's levied on the point of sale of a property and generates "upfront receipts" for the Treasury.
A land value tax, on the other hand, will be "highly bureaucratic" to implement and open to "significant challenge" from landowners because it is hard to determine the value of land separate to the buildings on it, he argues.
"For any new tax system you bring in, you would want it to be revenue neutral," he says.
"There is little doubt that [stamp duty] suppresses transaction activity. but it's about balancing that against the alternative," he says.
Proponents of an LVT say that by shifting the tax burden away from homebuyers, it will encourage retirees and empty nesters to downsize and frees up housing for growing families.
Cook says that implementing it on this basis would be tough politically.
Property is an "emotive issue" for many people, he says.
"Some of them would just find it unaffordable and people would feel forced to sell their home of which they have an emotional attachment.
"In general you do want to encourage a bit of downsizing in the market, but people want to feel like it's their choice."
Burnham too has spoken out against what he views as the "highly regressive" council tax.
Council tax is an annual property tax in the UK introduced in 1993 and used by local authorities to fund essential community services.
Properties in England are categorised into eight valuation bands, A to H, based on what it would have sold for in 1991. The percentages are set based on the funding gap between how much a council needs to spend and the amount of money they receive in central government grants.
It's described as regressive in nature because lower value properties, in bands A and B, pay higher rates of tax than properties in higher bands.
Many politicians and economists have called for council tax reform, describing the current system as outdated.
The Lib Dems have long called for reform, while the IFS has previously said that a reevaluation of council tax bands is "long overdue".
Chancellor Rachel Reeves announced a High Value Council Tax Surcharge of at least £2,500 on properties valued at more than £2m to come in from 2028. The tax will be levied on homeowners, rather than occupiers, and runs alongside council tax.
Burnham reportedly supports a proposal by campaign group Fairer Share, which which suggests a levy equivalent to 0.48% of a property's value.
But Cook argues that a tax like this would be heavily weighted on London and the south-east of England, where house prices are generally higher than the rest of the country.
"If you are living in Wandsworth, you may be thinking 'why am I paying for libraries in Wigan?'"
However, Adam, from the IFS, argues that whether or not money would be redistributed from London and the South East depends on whether the government changes the central grants it gives to local authorities.
The IFS argues that an efficient way to generate tax revenue would be some sort of combined land value and proportional property tax - to replace stamp duty and council tax.
"There is a case for taxing the land but also a case for taxing the building," Adam says. %!s()
