What's the metaphorical meaning of "Tone-deaf"?

Just as a literally tone-deaf person is unable to comprehend the differences between musical notes, a metaphorically tone-deaf person is unable to comprehend the different facets/nuances of a given situation. A statement such a person makes might also be described as tone-deaf.

It differs from words like ignorant because when you're calling someone ignorant, you're just calling attention to the fact that they do not know; whereas calling someone tone-deaf implies they're incapable of understanding.

It means something like "not being aware of the mood or sensitivities of your listeners".

In the linked case, a wealthy public figure is complaining about money problems, when lots of ordinary people are having genuine money problems. The opposite of being "in tune with" her listeners, you might say.

Used literally, tone-deaf denotes a kind of insensitivity, insensitivity to musical pitch (M-W). Used metaphorically as in the linked op-ed, it denotes a gross insensitivity to how a remark or complaint is likely to strike, and offend, an important segment of its likely audience. (And today the likelihood that a politician’s comment will reach an audience that he or she does not intend for it is higher than ever.)

In the usual sense, "tone deaf" refers to musical insensitivity.

In the metaphorical sense, "tone deaf" refers to social (or socio-economic) insensitivity.