Leaders in Greater Manchester have "unanimously opposed" being put into the highest category of the government's new three-tier system for localised COVID-19 restrictions, amid a bitter political row with Westminster.
A call between Greater Manchester leaders and Downing Street officials failed to reach an agreement on new coronavirus rules on Thursday.
And, after those inconclusive talks, Mr Burnham branded the Tier 3 plans as "flawed and unfair".
"They are asking us to gamble our residents' jobs, homes and businesses and a large chunk of our economy on a strategy that their own experts tell them might not work," he said.
"We would never sign up for that."
Under Tier 3 restrictions, people are banned from socialising with other households both indoors and in private gardens, while bars and pubs are closed unless they can operate as restaurants only.
However, Mr Burnham said Greater Manchester leaders had been told by England's deputy chief medical officer, Professor Jonathan Van-Tam, that "any regional lockdown would require widespread closures way beyond pubs to stand any real chance of working".
Professor Chris Whitty, England's chief medical officer, has previously admitted Tier 3 restrictions "will not be sufficient" to slow COVID-19 infections alone.
Businesses forced to close as a result of localised restrictions will, from 1 November, see the government pay two-thirds of their employees' wages.
But Mr Burnham said the government should be offering the same support it did through the furlough scheme, which will finish at the end of this month and has seen the government pay 80% of employees' wages.
He accused ministers of testing out the three-tier approach on parts of northern England.
"Greater Manchester, the Liverpool City Region and Lancashire are being set up as the canaries in the coalmine for an experimental regional lockdown strategy as an attempt to prevent the expense of what is truly needed," the mayor said.
"The very least they should be offering the people of Greater Manchester who will be affected by these closures is a full and fair 80% furlough for all affected workers, 80% income support for people who are self-employed, and a proper compensation scheme for businesses.
"So far, they have not been prepared to offer that."
Urging the government to pursue a different course and consider a return to stringent England-wide restrictions, the Greater Manchester mayor said: "I've said it may be that we need to look at a national circuit break as preferable to this unfunded, risky regional lockdown strategy.
"We have to protect the health of the nation but let's do it as one nation, and not make the north of England the sacrificial lamb for an ill-thought-through Downing Street policy which doesn't make sense in the real world."
Mr Burnham claimed Prof Van-Tam had told Greater Manchester leaders that "the only certain thing to work is a national lockdown".
"But the government told us this morning it is unwilling to do that because of the damage it will do to the national economy," he said.
"And yet that is what they want to impose on the North West."
Mr Burnham accused the government of "treating us with contempt", adding: "People are fed up of being treated in this way, the North is fed up of being pushed around.
"We aren't going to be pushed around anymore."
Health Secretary Matt Hancock called on local leaders to "set aside party politics" and back the government's plan for increased restrictions.
"The situation in the northwest of England is severe," he said.
"The number of cases is rising exponentially, the number of people in hospital has doubled in just 12 days.
"So I call upon local leaders to set aside this party politics and to work with us to put in place the measures that are needed in Greater Manchester, across the North West, so that we can deal with this virus and support people through it.
"This is a time for people to come together so that we can control this virus."
Sir Graham Brady, the chair of the influential 1922 Committee of Conservative MPs, warned the government that forcing Tier 3 restrictions on Greater Manchester would be "a very foolish thing to do".
The Altrincham and Sale West MP told Times Radio: "The danger is, if you try to do these things without consent, people lose patience very quickly.
"We have a very clear demonstration at the moment; you have no support among MPs, no support among council leaders and opposition from the mayor as well.
"There clearly isn't the broad consent for this measure that is needed."