Most FTL travel basically has the planet remain stationary in a bubble of space and then moves that bubble at FTL speeds through real space.
In this case the effects on the planet would be the same as for any other spacecraft - whatever the effects of the drive might be.
The main additional consequence though is that you have moved away from your star. Gravity would keep your atmosphere in place but you would have immediate night that never ends. Temperatures would start dropping quite fast and keep dropping, I've not done the maths but I expect the surface would start freezing within weeks although it would take time for the freeze to spread and geothermal activity would slow it in some places.
You also need to consider two more things, the first is the effect on the system the planet is removed from. Orbits could well be distorted or changed. For example depending on where it was in its cycle around us our Moon could well swing away from it's current orbit and go rogue were the Earth to disappear.
The second is the inertia of the planet. You put it into FTL with a certain velocity. When it emerges from FTL then does it keep that same velocity? If not what happens to it? You will somehow need to synchronize the velocity of the planet with a suitable orbital velocity for its new position.
One way around this might be to use the FTL drive to place you in a suitable place next to a gas giant and let the giant's gravity accelerate you. By pulsing the FTL drive to remain in the right place you could achieve a constant acceleration until the required velocity is achieved.
As a side note this could potentially be weaponized. It was suggested in E.E. Doc Smith's Lensman series that they could find two planets that at some point in their orbit had opposed directions. They attached FTL drives to both planets and then moved them into positions each side of the target and turned off the FTL drives at the same time, causing the two moved planets to pancake the target's planet between them, overwhelming all that planets defense systems completely.