Maseches Derech Eretz Zuta Ch. 5 states: "One who is a Talmid Chacham (Torah scholar) should not eat while standing." From the context it is evident that this behavior is considered unrefined. Apparently there would not be anything halachically wrong for a non-scholar to eat while standing, though it's quite understandable why one would want to avoid it. Certainly it is not forbidden to do so -- the Gemara in Berachos 51b actually discusses cases where one ate while traveling or standing. (I would note that regarding this matter, a "Talmid Chacham" is probably anyone who is perceived as a Torah scholar, even if he does not consider himself to be particularly learned.)
Another point is that the Gemara in Gittin (70b) lists eating (or drinking) while standing as something that damages a person's body. Rambam (Hilchos De'os 4:3) simply states that one should sit or lean to his left side when he eats.