The title company that closed a purchase transaction has contacted us claiming that they mistakenly overpaid the county treasurer for property taxes. The treasurer does not provide refunds in any instance. The title company is asking us to withdraw the funds from the borrower's escrow account and give it to them.

The escrow funds are the borrower's funds much like a deposit account. I wouldn't withdraw the money from the borrower's checking account without their approval for this purpose. If there was no escrow account, there would be no money for them to ask back.

It's their mistake, they need to deal with it. If they want to go back to the borrower and ask for it back they can do that but I don't think it's our responsibility to clean up for their mess. Am I wrong here?

Nope - their problem not yours.

And the title company has no right to go to YOUR borrower to ask for money they paid in error. You paid them to do a job. Someone there dropped the ball. Your borrower has no obligation to pay for the title company's error.