Based on reading "Dulce et Decorum Est", do you think the narrator would most likely defend or reject the idea of World War III?

First, we should distinguish between whether Owen would think a WWIII is likely to happen and whether he would think that England should participate in it. 

In response to the first question, "Dulce et Decorum Est" was written in response to the trench warfare of World War I, and thus does not speculate about even the possibility of a second, much less a third, world war. Owen himself died in 1918, in World War I, and thus has not actually written about any wars later than that date.

In the last lines of the poem, Owen argues that if you could see and hear injured and mutilated soldiers, you would not consider war "dulce et decorum". Thus he would definitely be opposed to entering into a war for motives of glory or national pride. In the case of World War II, Hitler was conquering Europe and attacking Britain. Surrender to Hitler meant making extreme moral compromises, such as allowing the extermination of Jews, gypsies, and gay people. Although Owen might have objected to the British intervention in Iraq, self-defense against Hitler wasn't really a matter of glory-seeking but of survival against an evil dictator. 

In the case of a hypothetical third world war, whether Owen would support entry into it or not would depend on the nature of the war. Although he would tend towards pacifism, not wanting soldiers to die horrible deaths, he might feel that wars of self-defense or wars against imminent threats are justified.