Q. Is she required to file any Ohio state or local tax return if she doesn't want any refund since it costs more to add a second state to TT?
A. No.
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.
Only if Ohio state tax is mistakenly withheld, do you have to file an Ohio return to get a refund. You are not required to do so. You can let Ohio keep the money.
However the reciprocity agreement does not apply to local city taxes. You are not required to file an Ohio local city return, but you are also not entitled to a refund of the OH city withholding (box19 on your W-2), unless there was an error in the amount (unusual).
Michigan will not give you a credit for Ohio state tax withheld. You have to file an OH return if you want a refund.
Michigan will give you credit, on the MI state return for OH city tax withheld, 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 would have to file one to get that MI credit, even though there will be no tax due or refunded from the Ohio city. MI does not require you to add that credit to your MI return.
