Can you take an Immortal Phoenix out of the game?
'Dies' means 'is put into the graveyard from the battlefield', so any effect that destroys it will end up causing this trigger to happen. The four main ways this card could move from the battlefield to the graveyard is:


Being dealt lethal damage
Having toughness reduced to zero
Effects that say destroy
Sacrificing the creature.


A creature going to the graveyard and removing a creature from the game* are very different things, but there are ways of putting Immortal Phoenix into the graveyard (and keeping it there) and exiling it.

A card like Oblivion Ring can exile it, and some cards like Annihilating Fire can change where it goes. (If it goes into exile instead of the graveyard, it never actually 'died')

If Immortal Phoenix is put into the graveyard from anywhere that's not the battlefield (perhaps by being discarded from Mind Rot), that's also not dying, so it does not return to your hand.

One more complicated way of stopping Immortal Phoenix from going back to a hand is by preventing the creature's ability from happening. You could either counter the ability (such as with Disallow) or by causing the ability to not trigger, such as with Hushbringer

*This is called 'exiling', although there are technically ways to get it back from exile