Sarah Boyack, Scottish Labour's local government spokeswoman, said capping council tax bills at 2007 levels had "broken" council funding.
She urged Mr Swinney to bring forward plans for a "fairer" system which would also help plug growing gaps in council budgets.
She spoke out after Solace, the Society of Local Authority Chief Executives, told MSPs that freezing bills for the next two years - as promised by the Scottish Government - would be challenging and could only be achieved alongside significant cuts.
In a separate warning, the Accounts Commission released a report detailing sharp rises in council charges for services such as adult social care, sports and leisure, planning consents and car parking.
Charges reached £1.3 billion last year, equivalent to 57% of the cash raised from council tax. A decade ago charges amounted to 40% of council tax revenue, the public spending watchdog found.
Ms Boyack said: "The SNP's underfunded council tax freeze has broken local government finance.
"Coupled with cuts to local authority grants, the freeze has placed a huge burden on councils who have to make increasingly difficult decisions on which frontline services to cut.
"John Swinney needs to stop obsessing over independence and tell us now what his plan is to put in place fairer funding for local government."
She claimed many people were paying more in increased council charges than they were saving as a result of frozen tax bills.
Some of the most vulnerable were also being hit as vital council services were cut back, she said.
The call will reopen the debate on the future of local government finance, seven years after council tax bills were frozen by the incoming SNP administration.
The measure was intended as a stop-gap until the SNP's promised local income tax was introduced.
However, the plan was shelved following widespread opposition, despite SNP claims it would reduce the burden on those on average and low incomes.
A report by the independent Institute for Fiscal Studies think tank earlier this week said property taxation in Scotland was "ripe for reform".
It added: "The failure of successive Scottish governments to date to undertake even the most obviously desirable reform to council tax - a revaluation to base it on up-to-date property values - does not bode well for making politically difficult improvements under independence."
Despite the SNP's support for a local income tax it also argued an internationally open, independent Scotland should prioritise property taxes, which are harder to evade, rather than relying solely on taxing income or spending.
Ms Boyack said: "The SNP keep saying they will bring forward their plans 'later in this parliament' yet we are half way through the term and they have said nothing on how they plan to fund council services. We need to know what their plans are for local taxation, how they will work and how they will impact on household budgets."
A Scottish Government spokeswoman said freezing council tax had proved popular and saved the average household around £1,200 since 2007.
However, she added: "The Scottish Government considers the current council tax system as a whole to be unfair and as set out in 2011 we have committed to consult with others to produce a fairer local tax based on ability to pay later in this Parliament."