Can I use Forge of the Future's ability on my opponent's turn?

No, you cannot use the activated ability of Forge of the Future on your opponent's turn.

According to Rule 374, all activated abilities can only be activated on the controlling player's turn and during an open state. Additionally, Rule 148.2 specifies that the activated ability of Gear may only be executed during the controlling player's action phase during an open state, and not during a showdown.

As noted in the official FAQ for this card, "Forge of the Future does not activate when killed by an opponent's Adaptatron conquering a base because Forge of the Future is an activated ability, not a triggered ability." Furthermore, because it lacks the Reaction keyword, it cannot be activated in response to your opponent's actions.