When will TurboTax pay my Illinois taxes

Read the state payment instructions carefully, since most states cannot be paid from within the TT system, and thus require additional steps.

https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe

Your state tax due cannot be paid from your federal refund. 

It is not possible to pay your state tax due from your federal refund, nor is it possible to pay a federal tax due from a state refund.  The federal and state treasuries are completely separate from each other.  You may have seen a TurboTax summary of your federal and state amounts that confused you into thinking one was “paying” the other.  It is not.  If you have federal tax due you can pay by mailing your payment, having the payment taken out of a designated bank account, or you can pay directly on the IRS website.  You must pay your state tax due using the state’s preferred method of receiving payment.  For most states that will be by making a payment to the state’s own tax website, or by mailing a check or money order.

https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe

To check the IRS or state payment date selected in TurboTax:

1.     Sign In to TurboTax

2.     Select "Taxes"

3.     Click "My Tax Timeline"

4.     Choose "Some things You can Do"

5.     Select "Download all forms and worksheets"

6.     Review Part V of the Federal Information Worksheet for payment and banking information.

7.     Review State Information Worksheet for payment and banking information.

** If the check boxes are checked "NO" and your banking information is not displayed on the Information Worksheet, then the direct debit was not successful. 

To setup payments for federal and state taxes, use the links in this message.

If you have not set-up your payment through TurboTax , visit  www.irs.gov/Payments  and choose from the payment options available. 

That reply above wasn't specific enough. The software doesn't tell you how you paid your taxes. You need to go to the printed pdf version to find out your method of how you chose to pay. Then follow those instructions to make your payment.