Depends on exactly what you mean by “the weight of the base”, but if you mean “the value that a scale would read if the base were placed on it”, the answer is “yes, it increases by an amount equal to the weight of the levitating object”.

The simple reason for this is that there’s no physical distinction between holding something up by (say) a physical bar and holding it up by a magnetic field.

But if you want to do the math, let’s consider the forces involved. Positive is up, negative is down.

* The levitating object feels a downward force *-mg* from its weight (where *m* is the mass of the levitating object and *g* is the acceleration from gravity). Since it’s not accelerating, it must be experiencing zero net force, which requires an upward force *+mg* from the magnetic field.

* The magnetic base is applying a *+mg* force to the levitating object and therefore must, by Newton’s third law (aka conservation of momentum) be experiencing an equal and opposite *-mg* force. The base also experiences a downward force *-bg* (where *b* is the mass of the base) from its own weight. So the base is pressing down with a force *-mg + -bg* on the ground, which is precisely its own weight plus that of the levitating object.

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