How to reproduce this rainbow color feedback effect in After Effects?
Sorry, I don't have keyable footage under my hand, so I tried to do it different. But it will better look with keying:



I duplicated my layer couple of times
Moved every layer couple of frames forward
Decreased Optically value for every layer with same step, so values distributed from 100 till 0 (last one can be skipped)
Added to them Hue/Saturation effect with distribution of Master Hue across them.


It is manual approach, but it is not so difficult or time-consuming. 

I believe it also can be achieved with Particle system, when your footage serves as emitter. Some of particle systems effects have "color over life" value, which will create your "Rainbow effect".

Good luck!