If my opponent uses an Akshan to take my exhausted Heart of Dark Ice, when they take it, is it still exhausted?

Yes, the Heart of Dark Ice remains exhausted when your opponent takes control of it with Akshan.

According to the rules of Riftbound, control-change effects (like the one created by Akshan) change which player controls the object, but they do not change the state of the object itself. Since the gear was already exhausted when it was taken, it remains in that exhausted state under your opponent's control.

As a reminder, because the gear is now under your opponent's control, they will be the one who decides when to ready it (during their own turn) and they will be the one who can use its ability.