For the sake of the argument lets use a black hole so the time dilation effects are greater and the planet in question is orbiting close enough to have significant TD effects but far enough to not fall in. So, in this scenario one part of the planet would be closer to the BH and therefore would experience a slower passage of time relative to the part of the planet on the other side. Would this cause the planet to "roll on time" due to the time dilation effects?
Assume it is not tidally locked.
So we are to assume that the planet is not gravitationally locked for some period of time. This could possibly happen for a neutron star because it's gravitational bulging would be minimal owing to its structural integrity. We also have to assume that $t_{rot}\ll t_{orbit}$, because of $g^{tt}$ varies appreciably from one side of the planet to the other, then so does $g^{rr}$ and I'm not sure what kind of orbit that would be (presumably different than the particle orbit one finds in a GR exercise).
That being said I think different stories will be told by different observers. So let's focus on someone looking at the planet from a far away distance, he will see the far side of the planet rotating faster than the one close to the horizon. If this side was really close to the horizon, then the observer will basically see a very small percentage of the planet's mass stretched out on the far side rotating at normal speed, and most of the planets mass saturated on the other side almost motionless. So even though the speed at which land is moving on the far side is relatively fast, it would take a single point on the surface a very long time to complete one orbit. This asymmetric mass distribution I think will cause gravitational locking which is peculiar in that there was no gravitational bulging involved in the traditional sense of the word. So in other words this would imply that time dilation will be responsible somehow for slowing down the overall rotation.
