My favorite, which I unfortunately never dug into enough (/don't remember the solution), was a bug in some horrible pile of Javascript that only raised an exception _if the developer console was open in the browser_. So on a normal refresh, everything goes fine, but as soon as you open the console to diagnose or inspect something else, bam, now you've got _two_ issues...
LOL, my current favorite is the one where our site session dies once the Dev Tools are opened in IE11 (but only on Windows 10). It makes debugging a serious pain.