also: ききすてならない
expression
inexcusable; unpardonable; can't be ignored
Used when something said is so offensive or unacceptable that one cannot let it pass without comment. Often follows a statement that the speaker finds outrageous.
彼の失礼な発言は聞き捨てならない。
His rude remark is inexcusable; I can't let it pass.
その冗談は聞き捨てならない。
That joke is unpardonable; I can't just ignore it.
許せない means 'unforgivable' in a general sense, while 聞き捨てならない specifically refers to something heard that cannot be overlooked.
From 聞き (hearing) + 捨て (throwing away, discarding) + ならない (cannot be). Literally 'cannot be heard and thrown away', i.e., cannot be ignored.