Lost Auramancers does not sacrifice when the final time counter is removed by anything other than the vanishing ability.

Lost Auramancers enters the battlefield with three time counters on it. The vanishing ability reads: "This creature enters the battlefield with three time counters on it. At the beginning of your upkeep, remove a time counter from it. When the last is removed, sacrifice it."

When the final time counter is removed via the upkeep trigger, everything works perfectly, however if Lost Auramancers is targeted by effects such as Vampire Hexmage, Heartless Act, or Price of Betrayal, Lost Auramancer