Why does Tim O'Brien, the narrator, consider himself a coward?

This is because he originally intended to run away to Canada but lost his courage during a six day stay in a lodge near the border. To Tim, actually making it to Canada and not fighting would have been mor courageous.

In "On the Rainy River," O'Brien tells why he, or should I say why his fictional character actually went to Vietnam. He almost went to Canada instead. His reason for going to Vietnam rather than Canada is not what you think though. What stopped him, ironically, was fear. "All those eyes on me," he writes, "and I couldn't risk the embarrassment. . . . I couldn't endure the mockery, or the disgrace, or the patriotic ridicule. . . . I was a coward. I went to the war."

Obrien considered himself a coward because he WENT to war.