Actually, O'Brien opens his story with the sentence, "This is true," which sets the tone for O'Brien's commentary on the nature of a war story and, perhaps more important, opens the story to metafiction.  For O'Brien's purposes, it is important that the reader set aside his doubts for  a moment and O'Brien's comment that the story is "true" creates what is known in literature as a "willing suspension of disbelief."  At the same time, however, the reader must be aware that if the author labels one thing as "true," there may be something in the story that is not "true"--literal truth, then, may not be absolute truth.

As soon as O'Brien labels the story as true, he has stepped outside the narrative into the world of metafiction--defined generally as a story that comments upon itself and, in the case of "How To Tell A True War Story," tells us exactly how to read the story by instructing the reader in the nature of truth about the war experience of Vietnam.  For example, after the first section of narrative about Rat Kiley's letter to Curt Lemon's sister, O'Brien tells us

A true war story is never moral.  It does not instruct nor encourage virtue, nor suggest models of proper human behavior, nor restrain men from doing the things they have always done.  If a story seems moral, do not believe it.

This comment quite accurately reflects what happens to Rat Kiley's letter to Curt Lemon's sister.  The letter is filled with what Kiley believes are Lemon's strengths as a soldier--blowing up villages, calling in artillery, fishing with hand grenades--but the letter doesn't get a response from Curt Lemon's sister--probably because she is horrified by the brutal soldier her big brother has become.  In other words, Rat Kiley tells a truth about his friend Curt that cannot be accepted by Curt's sister.

The tension between literal truth and Vietnam's truth is at the heart of the story.  For example, when the LRRP hears "strange gook music" in the middle of the jungle--a cocktail party, opera, full-piece orchestra--we know that event though that is not possible, we also recognize that, but this point in the story, literal truth may be less important than experienced truth.  In essence, if the patrol thought they heard music, music is what they heard.  At the end of this episode, Sanders confirms that even though this really didn't happened, it seemed to have happened.  And during one of O'Brien's explanations, he tells readers that, in Vietnam, what seemed to happen was more important that what actually happened--truth is felt in the stomach, not understood in the brain.

When the story starts with "This is true," we are confronted from that point on with what constitutes the nature of truth in war.  The highest evolution of the truth of a war story comes at the very end where we read

. . . a war story is never about was.  It's about love and memory.  It's about sorrow.  It's about sisters who never write back and people who never listen.

The opening line of Tim O'Brien's "How to Tell a True War Story" is "In a true war story, if there's a moral at all, it's like the thread that makes the cloth." The text does not, even in the first paragraph state "this is true." In fact, the text openly questions what a true war story will, and can, do to the reader. 

That said, if the text did begin with the words "this is true" its implicit (implied) problem would be the fact that the author, or narrator, felt it necessary to define the story as being true. For some readers, directly stating that something is true may make the reader question the truthfulness of the statement. Edgar Allan Poe and his unreliable narrators are infamous for this. For example, the narrator in Poe's "The Tell-Tale Heart" repeatedly states that he is not mad (or asks how the reader can call him mad). This forces the reader to question the narrator's mental stability. Perhaps if he had not brought it up, the reader may never have questioned his mental stability. 

In the same way as Poe forces certain questions to be asked of his narrator and text, by instantly stating that the story is true forces readers to question if it is true at all. In a sense, by stating that something is true, some people may look for any falsehoods in order to make a liar out if the speaker.