The phrase "it's no skin off my nose/teeth" is generally used to mean that something isn't much of a risk/concern. But where does it come from? Specifically with respect to teeth. What is tooth skin?
Learn English – Where does the phrase “No skin off the teeth/nose” come from
etymologyfigures-of-speechslang
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Best Answer
"No skin off my teeth" is not an expression in the UK. Apparently it does exist in the US where it means "no skin off my nose".
Saying "X is no skin off my nose" means that X occurring won't affect you in a positive or negative manner, the outcome will be neutral to you. The phrase is explained by phrase finder as being of boxing origin. I assume because boxers' noses are the body part most prone to damage.
As James McLeod points out "skin of my teeth" means only just or barely. This question addresses its origin.