Unless somebody can provide me with a situation where it would happen, I'm going to say no. Assume for a moment that it could happen. Said mountain would be in a case of unstable equilibrium, that is, one small nudge and it would go to either the moon or the planet.
The thing is, gravity is attractive and gets weaker the further away the object is. We're not talking about something similar to a magnet being suspended above another magnet. In that case, you have a repulsive force (the magnet suspending the other one) and an attractive force (gravity). Gravity pulls the top magnet down, it gets closer to the other magnet, the repulsive force grows, magnet goes back up, repulsive force weakens. The only thing that comes to mind that could possibly work would be the moon getting closer and further from the planet, changing the relative distance to the mountain, but I highly doubt that the mountain would take long enough to fall for that to change anything. I doubt it enough that until somebody smarter than me provides a proof with numbers, I will maintain that floating mountains don't work.