Since the withholdings were taken out for PA instead of OH, you will owe state income taxes on your OH resident state income tax return. Also since the PA taxes were already sent to the PA Department of Revenue, your employer will not be able to refund your PA withholdings.
OH and
PA have what is called a state reciprocal agreement. This allows nonresidents
to not have nonresident state withholding taxes taken out for wages earned
in the reciprocal state.
If PA withholding were taken out of your wages, you
will not be able to claim a state tax credit in OH for these
PA withholdings. Instead, you will need to file a PA nonresident state
income tax return to claim a refund.
In
order to get the full refund of your PA state income tax withholdings, you will
need to file a nonresident PA return but report zero "0" income from
PA (even though you do have PA wage income). You need to file your PA return
this way because OH does not allow a credit for tax paid to a reciprocal state
on wages, salaries and commissions. If tax was withheld by a reciprocal state
(PA), you must file directly with the state for a refund of those taxes. You
must mail in this PA nonresident state income tax return and include your state
W-2 information. You will want to include an explanatory statement with your PA
return stating your situation (that your PA employer withheld PA taxes from
your wages in error due to a reciprocal agreement with OH).
To
receive an PA exemption, a nonresident
employee must file Form REV-420, Employee's Statement of Nonresidence in
Pennsylvania and Authorization to Withhold Other States' Income Tax.
Here
is additional information about filing in multiple states (select "see
more answer" to view the entire attachment)
https://ttlc.intuit.com/replies/3300797
