Hell no, he shouldn't go! (Yet.)
I have to say, I think Oliver is jumping the gun by calling for Gen. Stanley McChrystal to be fired. As Talking Points Memo's Justin Elliott points out, most of the negative comments about the Obama administration don't come from McChrystal, but from aides who are not named.
McChrystal does seem to have poked Biden on the record, but A) if anyone can understand putting one's foot in one's mouth, it should be Joe Biden; and B) his comment about Biden--"Who's that?"--doesn't seem to rise to the level of "contempt" and could just as easily be seen as a self-effacing reference to his earlier clash with Biden.
(On the other hand, one of the general's aides does say "Biden? Did you say: Bite Me?" which could be easily considered an Article 88 violation, which is when "(a)ny commissioned officer...uses contemptuous words against the President, the Vice President" or other, specific officials.)
As a former military public affairs guy, though, the worst part of the article is that McChrystal's aides piss on civilian leadership while talking about McChrystal's feelings. The "Boss" thinks this or the "Boss" felt that. That's some major Blue Falcon bullshit right there and those aides should be embarrassed. You don't speak for the general, dammit. I don't know whether the aides weren't reminded by public affairs staff that they're supposed to speak only for themselves when expressing opinions or, hanging out with a reporter for a "cool" publication, they forgot the guy was a journalist and were trying to one-up each other with edgy comments, but if McChrystal goes down, it will be because of his staffer's big mouths.
On the other hand, if the full article seems to show this is the command climate McChrystal has created and supports--one in which soldiers forget they serve at the pleasure of civilian leadership and feel comfortable in deriding that leadership in the press--then, yeah, he's gotta go. I just don't think the stories about the story have yet provided enough evidence to make that call.
Update: Grumpy, Droopy and Huckleberry say they're cool with McChrystal being fired.