Can rotating the Nikon D5200 command dial quickly damage it?

I don't know what specific model rotary wheel Nikon used in that camera, but moving it fast shouldn't cause any excessive wear.

These rotary wheels are usually just rather simple mechanical switches. There are usually two separate switches. Each goes thru one complete cycle each detent, but the two are off from each other by 1/4 cycle. The fancy name for that is quadrature. With two quadrature signals, there are 4 states that get sequenced thru for each whole increment. It is quite easy to decode these two signals to keep track of the position of the wheel, regardless of what direction it is being moved in.

One part of the switch usually rubs against a part of the wheel so that it is moved into and out of contact with the other part of the switch. The wear on the device is therefore mostly related to total revolutions, not how fast those revolutions are, at least not for something you can do by hand.

If you spin it too fast, about the worst that is likely to happen is that the firmware gets confused and looses track of where the wheel is. It may miss increments, or even think you are spinning it the other direction.

Most likely the rotaty wheel will outlast your ability to spin it, whether you spin it fast or slow.