In which state do I pay taxes when I live abroad but work in the US?

@achall , generally IRS recognizes the source of income as where the work is performed.  Thus :

(a) if the work you perform for an entity in IL,  with a Tax Home  in say Germany, then the income is foreign sourced  ( Germany ) for US purposes.

(b) Many states  tax you on world income while a resident of that state.  Therefore  and when I was an Expat in Germany, many of my colleagues moved their residency , prior to  moving to Texas  or similar states with NO income tax -- just a driver's license  and a PO Box.  Different states deal with this situation differently.  

(c)  Please see the following information  ( including Pub 54 )  for Expats :

U.S. citizens and resident aliens abroad | Internal Revenue Service (irs.gov)

Is there more I can do for you ?

Illinois does not tax non-resident remote workers.  So you'll owe no taxes to Illinois unless you physically perform work within Illinois during the tax year..

Michigan will regard you as a resident (making all your income taxable by Michigan) unless you change your domicile to another state or country.  Michigan has a form which will help you determine this:

https://www.michigan.gov/taxes/-/media/Project/Websites/taxes/Forms/All-Years/3799.pdf