First off, don’t ever do it! From my experience there are several different reasons that can cause neck pain. Of the few times I’ve had patients sleep on the couch (even for a few hours), they end up with quite severe neck pain. They’re left in pain for weeks and it could easily have been avoided!
When people sleep on the couch, their head is usually elevated way too high and this causes a lot of compression in the joints of the neck. This compression leads spraining of the joints and causes the muscles around the area to go into spasm. In more severe cases, this can even lead to nerve compression and numbness or tingling going into the arm and fingers!
This can definitely be undone with a proper examination and treatment so there isn’t anything to worry about. Please remember to not sleep on the couch! If you do, make sure your neck is in a proper position that you’re accustomed to.