You cannot get Ohio city tax refunded. The city you work in is allowed to tax you and your employer is required to withhold that tax, just because you physically work there. It has nothing to do with your address.
You say you are not an Ohio Resident. Your home state may give you a credit for the Ohio city tax you paid. If you live in a city or county with an income tax, in you home state, they may also give you a credit.
You didn't mention OH state tax being withheld. If you are a resident of a reciprocal state
(MI, PA, KY, WV, IN), be advised that the reciprocity agreement does not apply to local city taxes.
