Oakham Town Council has been asked by Rutland County Council to maintain the monument, set to be installed outside the town's library on Catmose Street.
However councillors said they were worried about associated costs from potential damage.
The statue, in memory of the late monarch, is expected to cost £100,000 but is yet to be fully funded.
Chairing a meeting on the statue plans, Oakham mayor Sally-Anne Wadsworth said the town council had agreed to donate £5,000 towards the statue.
She said Oakham Town Council had also been asked to be responsible for the statue's insurance, cleaning, repairs and cost of removal if damaged.
Councillor David Romney said he was wary of taking on the responsibility, saying it could end up costing Oakham "an arm and a leg".
"We have no costs as to how long this lease is going to be, what it is for, and God knows how much it is going to cost when the kids start swinging on the arms and breaking the statue," he said.
The statue, was the idea of the Lord Lieutenant of Rutland, Dr Sarah Furness.
She previously said the statue will also include two corgis - one at the feet of the Queen and another at the base of the plinth for children to pat.
Councillor Paul Ainsley told the meeting: "I think we should remind ourselves that this is not Oakham's statue. It's Rutland's statue, and the initiative from the Lord Lieutenant was very clear that this was a statue for Rutland.
"We have no liability for this statue, and we should not assume liability for this statue - if this is for Rutland, then the best place for this to be financed is Rutland."
It is hoped the unveiling of the statue will happen in time for the first anniversary of the Queen's death and that King Charles III will attend.
Almost half of the required funds for the statue have been raised, but donations are still needed to pay for the project.