This is to keep local services running and help pay for social care reforms, the Institute for Fiscal Studies (IFS) think tank said.
It comes amid warnings that councils continue to face severe funding pressures due to the pandemic and must find new sources of income.
The government said it had given them £12bn since the start of the crisis.
Under current government spending plans, council tax bills will need to rise by at least 3.6% a year just to keep services running at pre-pandemic levels, the IFS said.
That would mean bills would have to rise by £160 by 2024-25, it said.
But extra cost pressures that eat into central government grants could easily push up council tax by 5% a year, or £220 by 2024-25, it said.
In addition, the government's plans for social care, which include capping costs, won't be completely paid for by a planned rise in National Insurance contributions, the IFS said.
The plans are likely to cost £5bn per year eventually, it said, nearly three times the funding currently allocated.
The government has made much of the idea that after decades of governments neglecting the increasingly pressing issue of underfunded social care, this was a nettle it was determined to grasp.
What's concerning in the IFS report is who might get stung.
A large part of the goal of the social care reforms was to address unfairness in the means-testing system.
But the IFS report highlights the risk that, without further funding, it could create new unfairness elsewhere.
To extend publicly-funded care to those who've spent £86,000 of their own money on fees will cost extra, and local authorities may only be able to recoup that by tightening eligibility criteria - so many poorer people lose access.
Second, the way the funds are allocated to local authorities dates back to 2013.
Because some areas like Blackpool have seen their populations fall, but others, like Tower Hamlets have seen them jump since then, the risk is that the money won't go where it needs to.
IFS research economist Kate Ogden said: "The government has stepped up with billions in additional funding for councils to support them through the last 18 months.
"It is likely to have to find billions more for councils over the next couple of years if they are to avoid cutting back on services, even if they increase council tax by 4% a year or more."
She said that the coming financial year is "likely to be especially tough", with ongoing Covid-19-related pressures and squeezes on budgets
She added that the local government funding system was "hopelessly out of date", being based on 2013 population levels, leading to "unfairness" in the distribution of resources between councils.
Council tax goes towards funding local services such as policing, the fire service and street cleaning. The funding is topped up with grants from central government and business taxes.
However, these central government grants were slashed between 2009-10 and 2019-20, especially in areas such as public transport, housing and planning.
The Local Government Association said councils "continue to face severe funding and demand pressures that will stretch the local services our communities rely on to the limit".
"The significant financial pressures facing local services cannot be met by council tax income alone," said LGA chairman James Jamieson. "Councils are particularly alarmed that the government's solution for tackling social care's core existing pressures appears to be solely through the use of council tax, and the social care precept."
Mr Jamieson called for local services to be "the top priority" in the upcoming Spending Review.
A government spokesperson said: "The government has allocated more than £12bn directly to councils since the start of the pandemic - with more than £6bn available to spend as they see fit - recognising that councils are best placed to deal with local issues.
"We have also taken historic action to fix the social care crisis - the Health and Social Care Levy will raise £12bn a year to fund the NHS and social care.
"The Spending Review will continue to focus on supporting jobs and delivering the public's key priorities."