He doesn’t really do pity.
In a sentence like that one, what's the difference between "doesn't pity" and "doesn't do pity"?
The verb pity is transitive: it requires an object:
I pity the people who live on the streets
So, if you want to say that somebody never experiences the feeling of pity, you cannot simply say
He doesn't pity
It's grammatically incorrect because it lacks an object: you would have to say
He doesn't pity anybody or anything
Alternatively you can use pity as a noun:
He doesn't experience pity
experience is a bit formal, but you can use do instead, in the sense to perform, take part in, or achieve something
He doesn't do pity.
The first sentence "he doesn't pity" is grammatically incorrect as it lacks an object: the second sentence "he doesn't do pity" is correct and is currently a fashionable way of expressing ideas of this kind.
In recent years it has become a kind of "hipsterish" thing to say that you don't do {noun}
. The phrase is deliberately ungrammatical. It labels the {noun} role (broadly construed) as with a rubric and thereby belittles it by turning it into a kind of stereotypical situation. The locution means that you steer well clear of {noun} because your temperament is averse to {noun}. Doing {noun} is not at all "who you are". You have no inclination whatsoever to be someone who does {noun}.
I don't do jealous girlfriend.
I don't do pity.
I don't do water-cooler.
If the expression is used of another person:
He doesn't do pity.
it is an ironic way of saying that it is not in his nature to take pity.