If I move X from one hand to another using his tap ability, when does he untap?
The tap symbol in X has no significant meaning while X is in a player's hand.

This is based on the comprehensive rules dealing with "tapping":


  701.19. Tap and Untap
  
  
    701.19a To tap a permanent, turn it sideways from an upright position. Only untapped permanents can be tapped.
    
    701.19b To untap a permanent, rotate it back to the upright position from a sideways position. Only tapped permanents can be untapped.
  
  
  107.5. The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since his or her most recent turn began. See rule 302.6.


A couple things here. Note that the idea of tapping as a general concept only applies to permanents. There is no reason to think from the comprehensive rules that a non-permanent can become tapped.

So how can you activate the ability? Well, you can still pay the cost of "tapping it", it's just that paying that cost will have no effect on its state; it won't become tapped. So tapping a permanent causes it to become tapped, but tapping a card in your hand doesn't.

Also, even if it did somehow become tapped; there's nothing in the rules from tapping it again to pay the cost... 107.5, which prevents you from paying the tap cost if its tapped, only refers to tapped permanents.

Since X in a player's hand is not a permanent, the regular rules regarding it being tapped or untapped simply don't apply.

Finally, if it somehow were tapped, there's no reason in the rules to think that it would ever untap (until it becomes a new object by changing zones, or some spell/ability untaps it). The only thing that causes untapping normally is this rule:


  502.2. Second, the active player determines which permanents he or she controls will untap. Then he or she untaps them all simultaneously. This turn-based action doesn’t use the stack. Normally, all of a player’s permanents untap, but effects can keep one or more of a player’s permanents from untapping.


As this rule only applies to permanents, the card in your hand wouldn't get untapped during the untap step.

Considering that these cards were creating primarily for the humor of seeing what silly things can be done, it would not surprise me if one of the reasons for having "tap" in the cost of the ability is simply to add to the silliness of the idea of taping a card in someone's hand. Though that is of course just speculation.