Ohio has a reciprocal agreement, on wages, with all neighboring states. No Ohio state taxes are withheld or due and you do not need to file an OH return. But your home state will tax you on your Ohio income.
If Ohio taxes are mistakenly withheld, you have to file an Ohio return to get a refund. You should ask your employer to stop withholding so you don't have to file an Ohio return every year.
However the reciprocity agreement does not
apply to local city taxes. You are not required to file a local city return,
but you are also not entitled to a refund of the city withholding (box19 on your W-2), unless there was an error in the amount. Michigan will give you credit for that on the state return,
but you have to attach a copy of the city return. So, even though the Ohio
city does not require you to file a return, you have to file one to get the MI
credit, even though there will be no tax due or refunded from the Ohio city.