No. You will not get a refund from Delaware. You will file a non-resident income tax form with that state and assuming your withholding was set up correctly, the amount you paid should roughly match up with what your tax liability to DE is. (Any dividends and interests you have will be associated with your state of residence, i.e. PA, and wouldn't really affect your DE tax liability.) Besides filing a non-resident DE state tax return, you will also need to file a PA state tax return and during which, you will get a credit for the taxes you paid to Delaware. That is the simple answer.
Here is one complication though. Even though Delaware's tax rates are generally higher than PA's state tax rate of 3.07%, you may still owe some taxes to PA if you had a contribution to an employer 401(k). This is because Delaware follows federal law and excludes any pre-tax contributions from their definition of income, whereas the state of PA defines income comprehensively and includes all amount paid by your employer as taxable compensation, even if some of it was contributed to a 401(k). To give an example, say your DE taxable income is $100,000 and you contributed $10,000 to your 401(k). With those numbers, your DE tax liability is say, around $5500 and your PA tax liability on $100,000 income is $3070. You would get a credit of $3070 and not have to pay any PA tax on the $100,000 that DE has taxed. However, you would still owe 3.07% X $10,000 or $307 to PA because that $10,000 of 401(k) income was never taxed by DE but it is considered taxable income by PA. If you did not have any such 401(k) contributions, then this discussion does not apply to you.
