I'm going to assume that the Underworld Dreams is in play when The Great Aurora starts resolving as opposed to being put into play mid-resolution. In this situation then no, you will not take damage from an opponents Underworld Dreams because it will have been shuffled into their library before you draw any cards. If you had a Banishing Light exiling an opponents Underworld Dreams, then you will take damage when The Great Aurora resolves. This is because exiling with a duration doesn't use the stack to return the permanent it will return to play mid-resolution, when all permanents are shuffled but before you move to the next step and draw.
Thanks for the hasty response.