Does the gear Heart of Dark Ice enters on my turn exhausted or ready to exhaust it?

Yes — Heart of Dark Ice enters play ready, like every Gear (Rule 149.1), so you can use its [T] ability the same turn you play it.

One timing caveat: it's a Gear activated ability, not an Action spell, so it can only be activated during your Main Phase in an Open State — not during a Showdown, and not on your opponent's turn (Rule 151.2). If you play it at a Base with nothing else happening, you tap it right away for the +3 Might.