In 2012 my employer withheld taxes to the wrong state. They paid withholding to Missouri instead of Illinois. When I filed in 2013, I filed my Illinois State Tax Return and paid all state taxes that were due. I then filed a Missouri State Tax Return return and received a full refund of all the taxes that were erroneously paid to Missouri. My question is, do I have to claim the state tax refund from Missouri as taxable income on my 2013 federal return?

Yes, if you ITEMIZED your deductions on your 2012 tax return.

If you itemize for 2013, you claim the Illinois taxes paid on Schedule A as well.