Why does Tim O'Brien (as the writer) "kill" Ted Lavender and not any other character? What is significant about his death?

I'm not sure
what you mean by "kill."  O'Brien doesn't use the active verb "kill."  It's
the passive verb phrase "was shot," "was dead," and "was shot and
killed."  All of it is in passive voice.

The first mention of Lavender's
death is on page 2:

Ted Lavender, who was scared, carried
tranquilizers until he was shot in the head outside the village of
Than Khe in mid-April.

The death is described
matter-of-factly, in passive voice.  O'Brien describes the death as if it were an everyday
occurrence, like the rain or a new mission.  It's part of war.  It's something else to
carry.

There is no one physically responsible for the death in Alpha Company.
 We must assume that a VC sniper took Lavender out with a head shot.  Morally, all the men feel
guilty for it, Cross the most.

Later, the third-person omniscient narration
says on page 7:

...now Ted Lavender was
dead because he lover her so much and could not stop thinking about
her.

So, it is Lt. Jimmy Cross who feels the most guilt
for Lavender's death, not O'Brien.  He carries the weight of the dead body literally and
figuratively (on his conscience):

But Ted Lavender, who
was scared, carried 34 rounds when he was shot and
killed outside Than Khe, and he went down under an exceptional burden, more than
20 pounds of ammunition, plus the flak jacket and helmet and rations and water and toilet paper
and tranquilizers and all the rest, plus the unweighted fear.

O'Brien has Lavender killed to serve several purposes: Lavender is a symbol of
unweighed fear.  The function of Lavender is to be a doppelganger, a ghostly twin to haunt Cross
(a Christ-figure), the one who feels the most guilt for his death.  The novel begins with a
death here, much like Hamlet does with the Ghost, and Lavender's death
hangs over the story and novel as a whole.  Like the Ghost in Hamlet,
Lavender will keep reappearing.  A major motif, as you know, is the ghost in
.  Observe some of the titles of the other stories: "Ghost
Soldiers"; "Lives of the Dead"; "The Man I Killed."

The purpose of storytelling, according to O'Brien, is to bring the dead back to life
through memory: it is to resurrect ghosts.  That is both the painful and joyous task of a
writer: to turn a war story into a love story by honoring the dead.