The death of Timothy occurs in Chapter 15 of this excellent novel, and, as we read it, we see that Timothy died as a result of protecting Philip from the storm that ravages their island. In particular, he dies from injuries sustained when he ties himself and Philip to the palm tree so that he is facing the brunt of the storm and acting as a shield to save Philip from the violence of the tempest. When the storm passes, it is clear that Timothy is unconscious. After Philip manages to untie them both, he feels Timothy and makes a shocking discovery about the condition of his body:
Timothy had been cut to ribbons by the wind, which drove the rain and tiny grains of sand before it. It had flayed his back and hsi legs until there were very few places that weren't cut. He was bleeding, but there was nothing I could do to stop it.
Philip goes to sleep holding Timothy's hand, and when he awakes, he discovers that Timothy has actually died. We thus see that Timothy died, sacrificing himself for Philip so that he could live.