I reside in MO but have a rental in IL. I feel I entered everything correctly but it appears to think the rental is in MO and IL is not reflecting any taxes paid or owed.

The rental activity is required to be included in the IL return because the property is located there.  If the income is low or the rental had a loss, there may be no IL tax due.

Resident State :  All income worldwide is required to be reported on your resident state return.  Any money that is also taxed in a nonresident state allows the use of the "credit for taxes paid to another state" when you complete your resident state return.  The credit is the lower of the actual tax paid on the income in the nonresident state or the tax that would have been paid in your resident state on the same income. Your resident state does not want you to pay tax twice on the same income.  

In a case where the nonresident state tax is zero, there would be no credit on the resident state return.