No Armenian church records have survived from Smyrna (Izmir). Records have survived from Bolis (Constantinople/Istanbul) and a few other locations.
There is an interesting book on Smyrna that the National Association of Armenian Studies and Research have in their collection (NAASR - located in Massachusetts). It has numerous photographs.
There is another small pamphlet on Smyrna that I have, again very little information that would be of genealogical value. The pamphlet is entirely in Armenian.
There is a book by Teotig that discusses the priests and their fate during the Genocide. So that may be a place to look.
You will find that there is so little information available to Armenians because of the Genocide that you will grasp any piece of information, regardless of how small or seemingly meaningless to others.
One suggestion I have that is often overlooked by beginning researchers is to look for the records of relatives. For instance, say your grandfather never survived the Genocide and only your father arrived in the United States and maybe your parents were married prior to the Genocide in Ottoman territory. So, a marriage record which would supply your grandparents names is not available. But suppose that your father's brother did marry in the United States, then his marriage record would supply the names of your grandparents as well. Taken further, suppose your grandfather's brother were to have come to the United States and married here, then his marriage record would supply the names of your great grandparents.
Not only that, they may have family stories and traditions that were not passed down to you.