We don't need no wings to fly (here).
It means we indeed need wings to fly.
But hey now,don't know girl we both know what to do. (here)
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Maybe you are thinking that it's a double negative(?)
don't ... no ...
actually in this case it means wings are not necessary.
It's the same as
we don't need nothing we don't need anything
Wings are not necessary (to fly)
As long as you love me (I can fly without having wings)
Maybe because it's a love song?
In the video, the words in your second sentence were changed to
But hey now, don't know girl, we both know it's a cruel world.