Does your face hurt?

Q: Does your face hurt?
A: No? why?
Q: Because it is killing me

This is an insult, and also a joke that relies on expectation whiplash. The jokester starts by sounding concerned for the person that they are telling the joke to, however, when the person on the receiving end of this joke responds they are insulted.

The insult "It's killing me" is a colloquial phrase indicating that the anteceding thing is causing them distress, so it is assumed that the persons face is ugly enough to cause the jokester distress from seeing it.