I have been asked if it is necessary to log off while in an inn in Warcraft.  No, not at all.  That's actually part of the joke here.  The only benefit to staying in an inn, is that the longer you are in the inn, the more bonus XP you earn while adventuring once you return.  It's meant to symbolize being "rested", which is why your character icon turns to Zzzz's when you are in an inn.

COD's adherence to this minor benefit, while ignoring the pain of someone he could be helping, shows us either:
a) A fictional avatar in a fantasy world is more important to him than a real life person, or
b) He lives so much in Warcraft that he has trouble weighing aspects of the game against reality, or
c) its a metaphor for me playing Warcraft instead of drawing a whole new set of panels for the next comic.  :)