Walking the dog is, first and foremost, the act of taking a petdog on a walk as part of an exercise regimen. Other meanings or similar phrases:
Walk the dog[1] is a common yo-yotrick where the player does a sleeper and then allows this spinning yo-yo to roll along the ground as if it were a leashed animal.
Wag the Dog is the title of a 1997 film starring Robert DeNiro and Dustin Hoffman, about the creation of a fake war in order to divert public attention from a presidential sex scandal. The term has subsequently found its way into common political vernacular to describe similar situations. See Spin (public relations).