No, because Aang did not die while in the Avatar State in any of the episodes. It is said that the Avatar Cycle will be broken if the Avatar is killed while in the Avatar State, but he wasn't killed .In the episode "Crossroads of Destiny" Azula shot Aang while he was in the Avatar State. Initially, it appeared as though Aang had died, but Katara was able to revive him, and his Avatar Spirit, by using the water from the Spirit Oasis. However, the injury jeopardized his Avatar State.