What exactly happens if I listen to the voices in my head?

As more reports from various players who have chosen one side or the other have started to roll in (such as in discussion threads like this one), as well as accounts from players who have played through the main story multiple times with different choices, there doesn't seem to be any confirmed consequence to these choices at the current time. Or, if there is one, it's subtle enough that nobody's been able to identify it yet.

However, as Funcom has stated that they intend for certain choices you make in the game to have consequences later on, it's entirely possible that your choice will affect something in some future content, but that's just speculation.

The consequences of these choices are revealed when you complete the quest atop Orochi tower.

At the end of the story line you are given an ultimate skill (triggered with the 'Z' button) that produces a large AOE effect and causes you to sprout wings. If you have rejected the gift all three times, these wings are gold in color. If you accepted three times, they are black. A mix of acceptances and rejections produces gives you blue or purple wings. These are only cosmetic changes. The effect of the attack is the same regardless.