Do I need to file state taxes for Pennsylvania if I received tax exempt unemployment there until I moved to another state in September of that year or not at all?

Unemployment compensation  are not taxable by the Commonwealth of Pennsylvania and local governments. If that was the only income received while living in PA, then you don't need to file PA return. 

Every resident, part-year resident or nonresident individual must file a Pennsylvania Income Tax Return (PA-40) when he or she realizes income generating $1 or more in tax, even if no tax is due (e.g., when an employee receives compensation where tax is withheld). 

Brief Overview and Filing Requirements Who Must File