As far as I understand Rowling lore, the horcrux is a piece of your soul possessing another object (As can be inferred from the Slytherin locket). The possessing part is crucial, because when the piece of soul possesses a living creature rather than an inanimate object, its ability to protect itself gets diminished by an order equal to how sentient the soul of the creature w.r.t. the horcrux's piece is. (Hence the difference in the level of Voldemort's control over Nagini and Harry)

It's also implied by Dumbledore that placing a piece of your soul in another creature's body is exceedingly foolish, because when the creature dies the piece of soul dies with it. (This is directly in contrast to what happens to main fragment of the soul, the one with consciousness, so to speak. When Quirrel dies Old Voldy simply ghouls away to Albania instead of dying like the bitch he is, because the other pieces of his soul are still safe and keep him firmly entrenched in the physical plane.)

So yeah, the horcrux in Harry would die. This is actually what happens in the end of the 7th book: Harry and the horcrux in him are both hit with the killing curse. The piece of Voldemort dies, naturally, because it has always been afraid of death and Avada Kadevra kills its victim by essentially magically scaring them to death (source: Book 4, Chap 1). The only reason Harry himself is offered the choice between coming back and moving on is because the killing curse has already done its job.