Othello is a Turk culturally, but he is a Christian and in the play, is fighting for Christians. He believes that the Turkish fleet has been destroyed in accordance with God's will. When he starts to suspect Desdemona has been unfaithful to him, he becomes somewhat obsessed with the handkerchief and almost treats it as having magical powers, which would be very anti-Christian. This, of course, is the work of the devil, in the form of Iago in this play. There are some instances where Othello refers to Iago as a devil (calls him "cloven foot", "demi-devil" etc.) but Iago refers to Othello as the devil, often focusing on his blackness.
Othello believes killing his wife is justifiable before God because he believes she has been unfaithful, but this goes against Christianity which preaches love and forgiveness. Christianity would condemn Othello's murder of Desdemona because the Bible says "Vengeance is mine, says the Lord." In fact, in the book of Hosea in the Bible, Hosea is instructed by God to marry a harlot (Gomer). Israel has been unfaithful to God by following other gods and breaking the commandments which are the terms of the covenant, hence Gomer symbolizes Israel who violates the obligations of marriage to her husband. However, at the end of the book, God forgives Israel, and in no way does God even suggest that Hosea should kill Gomer. So Othello's killing Desdemona is not Christian.
When Othello learns that he has indeed killed his wife without reason, he realizes that he is condemned and calls for his own death.
My view is that Othello is a type of Adam in that he is a noble man brought down by sin, just like the rest of us. "All have sinned and come short of the Glory of God." The conflict of the play is who is the greater "devil" - Iago or Othello? I belive it is Iago because he does not repent, whereas Othello is moral and does repent. He is sorry for his sin. His killing of his wife is seen by some as "sacrificial", so Desdemona is not an Eve. She would be more like a Christ figure because she is innocent, yet she is killed. Even though she is a woman, she is the sacrificial lamb because she dies for her love.
I don't know that I personally agree with the statement that Othello "transcends" Christian theology because this would imply that he is better in some way. I do think Othello illustrates Christian theology, but he does not transcend it. He is brought down by sin, so how can he transcend anything? He does not rise above sin. The way the play ends is tragic because Desdemona dies and Othello has killed her for no reason. Othello succumbs to the devil, he does not follow God's instructions to "resist the devil and he will flee."
Others will have differing views, but this is how I see it.