"The plane is taking off soon!"
了 is not an indication of past tense, it only indicates a change of status; another major use is as a modal particle, as in this sentence ("…oh, finally").
Ok, but is the 了 particle necessary? Does the sentence 飞机快要起飞 sound unnatural to a native Chinese speaker?
Since no context is given, it is optional. For an exclamation sentence, it may sound rather abrupt without 了.
It could also be spoken by a serious soldier and sound more brief and straightforward: “飞机快要起飞！无关人员请离开。The plane will take off soon! Irrelevant staff please leave this zone.”
I debated it for a moment, left 了 off and got it marked as incorrect. I suppose the sentence does sound just a bit incomplete without it, or perhaps even rude, as Learning German says. When to use and not to use "了" is just one of those subtleties of the Chinese language that has to be learned gradually. It's not used simply to denote past tense alone; it's far more complicated than that. "A change in modality" is a very good way to express it.
The le is necessary in the speech because it is often used when making a declarative statement of some kind. For example, "I am going home" will often be translated as "wo yao hui jia le."
This question is not incorrect by insisting on the addition of the word "le"