Does the power needed to raise a box onto a platform depend on how fast it is raised?

Yes, the power does needed to raise a box onto a platform depend on how fast it is raised.
Power delivered (P) by the body is equal to the force acting on the body (F) times speed of the body (v).
So, P = Fv …… (1)
The power (P) due to the force acting on a body is equal to the work done on the body (W) divided by time (t).
So, P = W/t …… (2)
When you raise a box onto a platform from the stationary position, if it take less time, then the speed of the box is more resulting more power is needed to raise the box from their original position. Therefore the power does needed to raise a box onto a platform depend on how fast it is raised.