the "mirror neuron" is one of these neuroscience facts that has a whole mythology built up around it, way more speculation and hypothesis and.. conceptual confusion.. than actual grounded facts and evidence.

we know that there are certain neurons that activate both when the animal performs some motor act, and when it sees the same/similar act performed. these are the prototypical mirror neurons discovered by Rizzolati and Gallese (edited to correct) et al back in the 80s and 90s. these seem to be a distinct class of neurons that are not in themselves necessarily part of the basic mechanisms for performing those motor acts.

later, in humans, there have been fMRI results showing that the same/similar areas (certainly not at the resolution of "neurons", which fMRI cannot reveal) are activated by e.g. being-in-pain and by seeing-pain, or by being-disgusted and seeing-disgust. the "mirror neuron" language was then extended to these observations, though it's not been demonstrated that they are in fact similar phenomena.

in fact i tend to doubt it. think about the fact that many acts or cognitive/emotional states might be a bit invariant to "who has it", especially when it comes to recognition or comprehension. like speaking a word versus hearing a word: in both cases, surely there are common mechanisms engaged that have to do with comprehending the word. But it's not a matter of "mirroring" the word somehow, at least not in the sense of mirror neurons as they were classically described.

with that in mind, sure: if we damage your "disgust" center in the brain, you might not be able to recognize disgust in others, but you might also be unable to experience disgust. so it's not that you've lost empathy - it's that you are now "agnosic" for disgust. it has nothing really to do with "mirror neurons" or any similar concept - it has to do with the ability to recognize or represent certain states or phenomena.

So with pain, similarly, perhaps if you lose the ability to experience pain you might become less sympathetic to others being-in-pain (since you can't imagine what that state is like). but again, it doesn't necessarily have anything to do with a "mirror system for pain", which may not even be a distinct thing that exists in the brain.