Does Ivern lose his tag after dying?

Yes, when a unit like Ivern, Friend to All dies and moves to the trash, it loses all temporary modifications and characteristics associated with its presence on the board.

According to Rule 110: "Whenever a Game Object changes zones to or from a Non-Board Zone, all Temporary Modifications of all kinds cease to be tracked on it in all capacities."

Because the tag gained by Ivern's ability is a modification to the card's state while it is in play, that tag is removed once the card leaves the board and enters the trash.