Hmm... songs that make me sad but shouldn't... okay, these don't always make me sad, but they're more likely to depending on my mood than any other song:
Streets of Philadelphia - Bruce Springsteen
Misty Morning, Albert Bridge and A Rainy Night in SoHo - The Pogues
Enjoy the Silence - Depeche Mode
Holy Grail - Hunters & Collectors
No Milk Today - Herman's Hermits
Glycerine - Bush
My Way - Frank Sinatra
Silent All These Years - Tori Amos
Great Southern Land - Icehouse
Old Irish Blessing - The Sprague Choir
Konstantine - Something Corporate
Walk Away - Bree Sharp
400 Miles from Darwin - The Whitlams
Cut Here and Friday, I'm in Love - The Cure
Walk On - U2
Puff, the Magic Dragon - The Irish Rovers
Erskineville - The Gadflys and Where Are You - The Whitlams, but that's because they were sung by Andy Lewis in both bands and released after he died.
Um. Wow, that was a bit long.