In The Giver, the people in the community are completely isolated from the outside world. Even their minds and memories are shut off from any intrusion from outside. This isolation includes a dome over the community, which provides climate control and apparently, the means of keeping out any contagious diseases. This serves two purposes, first, to protect the people, from evil and disease in the outside world, as well as from adverse weather conditions, and second, to prevent people from being tempted to leave the community.
We are told on the very first page about the rule against airplanes flying over the community, and we also learn that pilots do fly near the community, to deliver supplies, "in the landing field across the river" (1), where children watch to see the unloading but cannot go anywhere near the plane.
We then learn that a plane that was clearly not a supply plane flew over the community about a year before the story opens. The community's response was fear and consternation, and the leaders of the community forced everyone to "go to the nearest building and stay there" (2). It was ascertained that a pilot who was in training had made a mistake in flying over the community, so the people can resume their lives, and we also learn that the pilot was to be "released." What this means, we come to understand, is that the pilot will be put to death.
What can we infer from this knowledge? First, any plane flying over the community might be somehow spying on the community. Second, a plane flying over the community causes people to be afraid, which is not good for the community. Third, since the community has a dome protecting it, if a plane were to land or crash, it might harm the dome, exposing the people in the community to disease, destroying the climate control. Finally, if a plane were to crash and the pilot were to survive, the people in the community would be exposed to someone who might provide them with knowledge of that world, tempting them to leave.
A plane flying over the community thus provides exposure that could harm the community as well as the possibility of temptation. The survival of the community is dependent on strict isolation from everything outside.
The community in which the Giver is set is mean to be perfect. The reason Airplanes are not allowed to fly over the community is because with everything so uniform it becomes frightening or alarming when things are out of place.