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Aang has never killed a human being intentionally, due to his philosophy of all life being sacred. However, it has been debated whether or not he has taken human lives while in the uncontrolled Avatar State or when linked with La during the Siege of the North; for example, think about when he sunk several Fire Nation ships, leaving the soldiers to fend for themselves in the ice cold water.

Another debated point (although not about a human) is whether or not he actually killed the buzzard wasp that took Momo in a fit of rage. We know that he hit the creature with a powerful air swipe after which it fell out of the sky, crashing hard on the ground. We never saw the buzzard wasp again, so we can't tell whether he merely stunned it or actually killed it. Since he later did say to the spirit of Avatar Yangchen that he believes all life to be sacred and thus doesn't eat meat, one could make an argument for him merely having stunned the buzzard wasp rather than killing it.

While not shown in the animated series, Aang is known to have "killed" a spirit, namely General Old Iron: He struck the spirit in his chest in order to save Toph's life, after which the spirit dissipated and transformed into fish, retreating into the sea forever. As such, you could say that he didn't really kill the spirit as it lived on in another form, though it was definitely an action that weighed heavily on Aang.