I need a bank draft or money order (no personal checks or cash) for an
Irish citizenship application fee. The bank draft/money order must be
issued in Euros drawn on a bank in Ireland. How can I arrange for this
when I live in Michigan and do not have a bank account at an Irish
bank?

Go to your Michigan bank and ask for a "bank draft drawn on an Irish
bank in Euros."  The fees for a small check may be pretty high.

If the above suggestion does not work, I would contact the American,
assuming you are American, Consulate in Ireland.

I have tried asking my local banks. They can issue a draft in Euros,
but my banks do not have a branch in Ireland. I have posed my question
to the Irish Consulate and have been told that a bank draft from an
American bank is acceptabnle if that bank has a branch in Ireland, ie
Citibank. I have been unable to find such a bank that will issue a
bank draft to a non-customer. Thanks for your input.

Sorry, I should have also thought of money orders.

Here is the Western Union site.  

http://www.westernunion.ie/info/howToPickUpMT.asp

Good luck!