The restriction that the permanent must be tapped is checked only if you cast Dream Leash as a spell. Once Dream Leash is on the battlefield, it doesn't matter if the enchanted permanent becomes untapped.

If an effect puts Dream Leash onto the battlefield or moves it to another permanent, the permanent it becomes attached to need not be tapped. If you cast Copy Enchantment and copy Dream Leash as it enters the battlefield, the permanent it becomes attached to need not be tapped.