Google the Missouri Department of Revenue. - nm

You could call a tax preparation place, like H&R Block and they should be able to tell you. - nm

Info for you - Good Luck

I found this information (although I do not live in Missouri, I know it can happen to anyone, in any State, and was curious as to what needs to be done).  

If you have a good relationship with your employer, you should be able to tell them the following and have them take care of it immediately.  If not, you may have to go through the State of Missouri and have them send your employer an “official” letter telling them to reimburse you immediately.  

 “Reimburse your employee for the taxes withheld in error and file Forms MO-941C for each period affected.  When your forms are processed, you will receive overpayment notices advising to either use the credit on the next return filed or request a refund.”  

General Information Concerning Withholding Tax:
Missouri Department of Revenue
Division of Taxation 
P.O. Box 3375
Jefferson City, Missouri 65105-3375

Telephone: (573) 751-3505

E-mail: withholding@dor.mo.gov

http://dor.mo.gov/forms/4282_2011.pdf

Page 7, # 13

Overpayments:   Documentation supporting any overpayment must be submitted with Form MO-941C.  For example, if tax was withheld to Missouri and should have been withheld to another state, a W-2C must be provided.  If the adjustment is due to a calculation error, a payroll ledger showing the correction must be provided.  In other instances, the employer must provide a detailed explanation of the errors that occurred.  If the Department authorizes an overpayment, the overpayment may be used to offset a prior debt within your withholding tax account.  Any employer may review current withholding tax overpayments online at http://dor.mo.gov/business/creditinquiry/
From this site the employer can also request a refund.

Missouri E-Tax - KCMT

E-tax - Another MO MT

MO income tax - Not a MO Resident

If the company you work for is in Missouri, you might - have to pay Missouri taxes sm