Why aren't the characters named in "The Sniper"?

A good question that digs at the central themes of this story; in war, we strip the humanity away from ourselves and our enemies, in order to make the jobs of survival and killing easier. The ironic ending, and the reveal that the sniper's enemy was his brother, throws this abnormal existence into stark relief.

The morality in this story is all about proximity to the person being acted upon; if we were to ask the sniper, or anyone, "how do you feel about killing your brother?" they would probably say that this is wrong. However, take away that relationship, reduce your brother not just to an anonymous person, but to a person trying to kill you, and the response would probably change. De-humanizing one's enemies makes difficult choices easier; if the sniper had hesitated when shooting the old woman, thinking instead about how she had a name, a family, and so forth, then he would have decidedly put himself at a disadvantage, because she was doing things that would lead to his death. 

The sniper is probably not named because we are meant to reflect upon him as he reflects upon himself; he can't afford to be whoever he was before the war began, and has simplified himself down to his role; sniper. It might also be the author's choice to do this in order to encourage us to see the sniper as no different from any other combatant, and his depersonalization makes his choices easier to understand.

What is the significance of unnamed characters in "The Sniper"?

"The Sniper" is concerned with illuminating a universal condition of war.  In order to maximize this, O'Flaherty does not use names and specific identities to the characters.  Through this stylistic element, there is a universality that can be seen in the story.  The old woman who is killed in the crossfire could be anyone.  The two snipers are essentially people who are poised against one another in the charged atmosphere of civil war.  There is a universality in such a construction that is enhanced in not naming the characters.  They can apply to any context throughout the world, and not merely Ireland.  Ireland gives context to a universal condition in which people engage in war without reflection of the consequences until it is too late.  

In operating within a universal condition for his characters and their situation, O'Flaherty is able to make war a sad condition that all human beings endure and to which connection can be made.  The dehumanization that war causes is a universal condition that is enhanced by the lack of specific names in "The Sniper."