Well, I guess there are actually two peeves ... the first is when the expression is misspelled "here, here!"
The expression indicates hearty or vigorous assent or agreement with someone else's statement, and people usually get that part right. But the H-E-A-R spelling essentially implies that a vocal assertion has been made. "Hey, listen to what this guy is saying -- he's right!"
The expression is not supposed to indicate the position of something, like H-E-R-E. How much sense does that make? "I agree with this guy! YES! Right in this very spot where I'm sitting!"
So the second peeve, I guess, is when people use the expression as applied to a written assertion. How can you "hear hear" something that is text? Put your ear really close to the words on the monitor? Encourage others to do so because you agree with it?
I guess the peeve is really inclusive of a wide variety of idiomatic expressions that equate (or conflate) verbal communication with written communication. It's fairly common to refer to someone's text as what that person "said," or indicate that you look forward to "hearing" from someone when referring to an expected letter or message, and whereas these instances make my inner pedant wince, he dies a little every time "hear, hear!" is used.
[note: I am not advocating use, as an alternative, of the arguably more correct, but equally annoying, iawtc.]
This rant brought to you by the fact that I have to work all weekend.