in the same vein as run train, due to the fact that they are both sexual acts using transportation as a euphemism.
running plane consists of sticking ones fist in another, by way of vagina, anus, or anything else as far as it will go. using the same arm the fist is on, the instigator of the plane will proceed to lift the receiver up by whatever the fist is in and running around. if your lifting muscle is sub-par, the same can be accomplished by stepping up on a ladder and positioning oneself on top of said fist.
boy:im going to stick my fist in you
girl: i heard running plane is fun.
boy: ok i'll run plane on you
girl: weeeeeeeee this is soooooo fun, and sexy
boy: ooooh yeea