English equivalent of saying “Don’t get in between the nail and the flesh”?

You can consider the proverb "Don't go between the tree and the bark." (as common usages are listed already.)

It conveys the precise meaning you are looking for (analogy to nail and flesh) but it never gained a common usage; and might be mainly literary. [There are also other usages with the phrase "between the tree and the bark".]

Meaning: Don't interfere when two people are having an argument.

_{English proverbs / wikiquote.org}

It might be of European origin but it is mentioned in the novel the Modern Griselda by Maria Edgeworth (1804) which might be the first usage in English. [Maria Edgeworth is an Anglo-Irish writer].

But, an earlier usage is mentioned in Le Médecin malgré lui (a comedy by Molière first presented in 1666) where the character Sganarelle misquotes Cicero. (from the book The Dramatic Works of Molière by Molière)

Something that carries the same violence as the original might be the idea of getting in the crossfire, i.e.

Leave Bob and Alice alone! You don't want to get stuck in the crossfire, do you?