Thanks to zxc for clearing up what I meant to say (i.e., living body parts aren't "grammatically animate", even you might expect they'd be). And I like it-ogo's explanation that одушевленный derives from душа ("soul").

Я вижу вирус
Scientifically, this one makes sense, because a вирус isn't really "alive." (But, of course, a насекомое IS alive, so there's no real consistency.)

Regarding the animacy of мертвец but the inanimacy of труп -- perhaps the reason is connected to the fact that труп can mean "an animal carcass" in some contexts, but мертвец is used only for dead humans? (I'm just guessing -- because, of course, nouns for animals are usually одушевленный in a grammatical sense, regardless of whether animals are believed to possess a душа!)