If another player gains control of Dungeon Geists, its effect expires. It won’t keep the creature from untapping anymore, even if you later regain control of Dungeon Geists.

The ability can target a tapped creature. If the targeted creature is already tapped when it resolves, that creature just remains tapped and doesn’t untap during its controller’s next untap step.

Dungeon Geists’s triggered ability can target a tapped creature. If the targeted creature is already tapped when it resolves, that creature just remains tapped and won’t untap during its controller’s untap step.

If Dungeon Geists leaves the battlefield before its triggered ability has resolved, the target creature will be tapped, but it will be able to untap as normal.