This idiom actually has three variations:
“No skin off my nose.” “No skin off my teeth.” “No skin off my back.”
They all mean the same thing; something is of no concern and has little effect, if any. Example:
“It’s no skin off my nose if she wants to act that way.”
Where did it come from?
This wasn’t so easy to determine. I did a lot of digging on this one. Even though the meanings agree, the histories do not.
“No skin off my back,” is said to have been derived from the practice of whipping or flaying the skin off someone’s back as punishment and is primarily attributed to American use.
“No skin off my teeth,” is said to have come from the biblical mention in the Book of Job 19:20
“My bone cleaveth to my skin, and to my flesh, and I am escaped with the skin of my teeth”.
“No skin off my nose,” however, is a little bit more interesting. It’s said that back in 16th century England, a cult existed in a not so savory district much to the disdain of the King who was avowed to dismantle it. As a resort, the cult was forced to be more covert in their activities. It’s said in the places where they held their worship meetings a small hole was carved into the door. He or she who wished to enter had to place their nose into the hole. The worshipers would draw a small mole on the side of the nose as a sign they were followers. If there was no mark, the priest would take a knife and slice a strip of skin off the nose. The King’s guard quickly learned “not to stick their noses in places where it didn’t belong” because not only were the repercussions painful, they were obvious and earned a lot of teasing from the peasants. Therefore, they chose to “keep their noses clean.”