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Or would the planet just be subject to extremely intense tidal forces instead?

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No.

There are two simple arguments, each on its own alone enought to refute such thought:

First way: Tidal force is the gradient of the gravitational attraction. Gravitational force goes like $F_g \propto \frac{M}{r^2}$ and the tidal force thus as $F_t \propto \frac{M}{r^3}$. For a moon and its planet the mutual distance is the same. And as the planet has the higher mass, the tidal forces of the planet would be higher on the moon than the one of the moon onto the planet. Materials are not that different that it plays a major role here. So before a moon destroys a planet, the planet destroys the moon (or you mis-named the two in the first place, calling the planet the moon and vice versa). This may happen for moons within the Roche limit of a planet.

Second way to argue: The extreme case of 'many moons' is actually a ring of moons, thus a complete ring around the planet. However a spherical symetric mass assembly around a center of mass (here the planet) does not excert any tidal force anymore. Thus a circular mass assembly will only create a small gradient in the normal direction to the equatorial plane and increase the equatorial radius a bit.

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