The most famous suicide song is "Gloomy Sunday", known alternatively as the "Hungarian Suicide Song". Much urban myth has shrouded the song upon its creation in the early 1920's. It is credited with causing a rash of copycat suicides amongst its lovelorn audience when it was first released. Also famous is Fade to Black (song) by Metallica.
On the other hand, Billy Joel and R.E.M. each wrote a hit song "You're Only Human (Second Wind)" and "Everybody Hurts" respectively which were specifically written to dissuade people from contemplating suicide.