To what character (or characters) does the title refer? Is it used straightforwardly or ironically?

The title refers to Sergeant Nathan Marx. However, it is an ironic reference.
In the story, Marx's fellow Jew, Private Sheldon Grossbart, habitually manipulates Marx into doing him favors. Grossbart believes that their shared religious heritage is reason enough for him to receive special treatment from Marx.
The text tells us that Grossbart manages to get himself and two other Jewish soldiers (Fishbein and Halpern) excused from barrack cleanings on Friday nights (the start of the Jewish Sabbath). Essentially, Grossbart takes advantage of Marx's natural loyalty toward his fellow Jews in order to receive special treatment. To flatter Marx, Grossbart slyly addresses Marx as "Sir" whenever he can, despite the fact that Marx is not an officer.
The text tells us that Grossbart next complains about the non-kosher rations at camp. He makes such a scene that Captain Barrett (Marx's commanding officer) is eventually embroiled in the embarrassing affair. Meanwhile, Marx is put on the defensive and finds himself having to explain Jewish dietary laws to Barrett. In the end, after upsetting the peace with his complaint, Grossbart writes a letter (in his father's name) to Congressman Franconi, specifically thanking one Sergeant Nathan Marx for helping Private Sheldon Grossbart change his mind about the Jewish dietary laws. 
Meanwhile, a separate letter (also written by Grossbart) informs General Lyman (the post commander) that Sergeant Marx is a credit to both the Jewish people and the United States Army. Grossbart certainly does enough to try Marx's patience. He even wheedles his way into earning a pass for himself and his two friends to observe Passover, despite the fact that no passes are allowed soldiers during basic training. Of course, Grossbart, Halpern, and Fishbein are not really interested in celebrating Passover; they only want to see how far they can go in using their Jewish faith to flout Army policy.
Frustrated beyond endurance by Grossbart's antics, Marx decides to regain the upper hand. He manipulates matters so that Grossbart, Halpern, and Fishbein are reassigned to the Pacific, where the fighting is the fiercest. The reference to Marx as the "defender of the faith" is ironic because Marx ends up having to "betray" Grossbart in order to preserve the dignity of the Jewish faith.
Grossbart's conception of what Marx owes him is based on a narrow and self-serving definition of the term "defender of the faith." He believes that he is owed special favors from Marx because they are fellow Jews.
By allowing himself to be manipulated for Grossbart's gain, Marx is actually tarnishing the reputation of the Jewish religion. This is why Marx eventually arranges to have Grossbart, Halpern, and Fishbein reassigned to the Pacific. In reassigning the three (thereby removing the ugly stench of favoritism from the whole affair), Marx is able to restore the reputation and dignity of the Jewish religion. He has to perform what is considered a negative action in order to bring about a positive outcome. Therein lies the irony of the title.