I am Active Duty military and resident of New Mexico. I have not paid state taxes in years. Why is Turbo Tax saying I owe 101.00 in state taxes this year? Thanks

It depends.  You are correct that NM exempts the income.  You will need to check some entries to have the income exempted.  Do the following:

1. Login to your account and click on Take me to my return.  
2. In the Personal Information screen, make sure you state that you lived in New Mexico (and not the state where you are stationed.  Also make sure that where it asks you if you worked in another state other than New Mexico, that you also answer no (unless you had other income that is not active-duty you earned in another state.)
3. Click Edit on your W-2, and hit Continue at the end of the W-2 screen to the list of Uncommon Situations
4. Make sure that Active Duty Wages is selected (see the Screenshot below)
   

On the NM resident return, your Active-Duty income will be exempted from NM tax.  Here is the screenshot to assist you:

Thank you very much. I remember that screen, but I must have glossed over that box

You're welcome