In Marvel's The Defenders we can clearly see that the Iron Fist can damage Luke Cage's jaw. If he had broken his jaw, sliced some of his flesh, broken a tooth etc., could Luke Cage have gotten proper medical attention? Could an anaesthetic needle, stitches or a dental drill help him?
Is there any mention of something like this in the comics?
Not much would happen.
In JJ (Jessica Jones), Claire needed to reduce intracranial pressure because Jessica managed to injure him enough to knock him out. She had to go through the eye socket to get there.
EDIT:
In the Luke Cage series:
They showed that bathing him in the right kinds of boiling acid will loosen his cells enough to cut between them and pull an alien metal bullet out.
In the Luke Cage series, he is shot by a bullet created with Chitauri metal fragments that is able to pierce his skin, so presumably surgical equipment made with exotic materials such as this might suffice.
In any case, he undergoes surgery later in the series to remove said bullet fragments. To do this the original experiment that gave him his powers is recreated, and he is submerged in an acid bath to soften his skin enough to open it back up with a scalpel.
