What should I do if local taxes were withheld incorrectly?

Tell your employer to stop doing that.
File a local income tax return and claim your refund.

Tom Young

What if they put the wrong address and started taking out taxes for the wrong county? 

Get the employer to fix the error now before the end of the year. 

You, most likely, will have to file a county return for the wrong county, to get   a refund of withholding.  And file a county return to your home county to pay that one. 

For more specific advice, you need to identify the state and the counties.