The phrase ‘no skin off my nose’ means that the speaker does not care about something because it doesn’t affect them.
The first written reference of the phrase is from 1910 in ‘The Cosmopolitan: Volume 49’. There is a quite wonderful story, which is, sadly, almost certainly untrue, that a cult in the 16th century would test if visitors were friend or foe by requiring them to put their nose through a hole, and if the secret nose signal was ok, allowing them to enter, otherwise cutting off the nose.
More likely the origins are a bit less exciting and relate to travelling, the phrase was common in the early 20th century, cars were a rarity and most journeys were completed on foot or by horse, long journeys would have an effect on the traveller; the exposed face would feel the worst of it, in the sun being burnt and in the winter being exposed to wet and frost – the most prominant part was, of course, the nose and the skin of this would be most affected by the jouney. The idea being that if you had to get involved in something other than the normal homelife, you might have to travel, if it was far, you would lose skin off the nose but if it was only a short journey (and thus, not as much affort) then it was no skin off your nose.