If you care about the Arl's health, however, it's one you'll undertake. However, undertaking such a quest isn't mandatory. If you agree to find the sacred ashes you heard about earlier, which are said to cure any illness, then you will receive a new quest from Teagan called The Urn of Sacred Ashes.
Teagan is most thankful for your hard work in freeing Connor from the devil's grasp, though he laments that the Arl is still very much sick. Teagan has recovered from his brief possession, and the injuries sustained when you gave him a beat-down in battle. But when you find yourself returned from The Fade and back at Redcliffe Castle, the first person you're going to end up speaking with is your new friend, Teagan. With the devilish possessor of Connor slain, Connor is now back to normal.