When I destroy a face-up monster with Doom Dragon it inflicts the damage, however when I target a face-down monster it does not deal the damage. The card does not specify the monster to be destroyed has to be face-up, therefore it should inflict damage equal to half the ATK of a face-down monster as well right?

It has been ruled in both the TCG and OCG that Infernity Doom Dragon will not inflict damage if it destroys a face-down monster.

This specific question is already answered in the rulings section for this card.

The damage dealt is half of the ATK of the monster while it was on the field. If you destroy a face-down monster, no damage is dealt.

http://yugioh.wikia.com/...gs:Infernity_Doom_Dragon 

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