Does Ivern lose his tag when he is being put in the graveyard?

Yes, when Ivern, Friend to All moves to the trash (your graveyard), he loses the additional tag he gained.

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. This is also confirmed in RiftJudge FAQ #9364 and FAQ #9754.