No, it doesn't work that way, thats a typical misunderstanding of the way it works because you're confusing observations from your perspective with reality.
The dot isn't moving at all, and you wouldn't actually have a dot painted on the surface of the moon. From your perspective on Earth it may appear that way, but appearances are often deceiving.
What you end up with is something that would resemble a dimmer (than the dot would be if stationary) line or blur on the surface of the moon, spread out over vast distances, that would appear to the observer on the Earth as though it was moving faster than it actually is. When the reflection of the light from the laser returned to your eye, the line would appear as a dot again, all due to perspective of the viewer.