phrases - A formal way of saying 'rub it in'.

I suspect that "don't rub it in" is an allusion to rubbing salt into a person's wounds, to make them hurt more. Christine Ammer, The Facts on File Dictionary of Clichés, second edition (2006) confirms that the connection is likely:

rub it in, to To stress something unpleasant or annoying in a teasing way; to ADD INSULT TO INJURY. The it in this expression may well be the salt that is in the much older related term, to rub salt into a wound, which dates from late medieval times (or earlier) and is still current. Rubbing it in originated in America; T.A. Burke used it in 1851 (Polly Peaseblossom's Wedding): "When it comes to rubbin' it in, I always ... roars up." Also related is the cliché to rub one's nose in it, meaning to reminfd one of a humiliating error or experience. [Citation omitted.] It alludes to rubbing a dog's nose into a mess it has made.

A formal way to refer to conduct that has an effect similar to that of having salt rubbed into one's wounds is to say that the conduct "exacerbates the original injury [or embarrassment]."

According to Merriam-Webster's Eleventh Collegiate Dictionary (2203), exacerbate means

to make more violent, bitter, or severe {the proposed shutdown ... would exacerbate unemployment problems —Science}