"Come As You Are", with the lyrics "No, I don't have a gun". Oh Kurt...
Civil Twilight's cover cranks it Up to Eleven. From the instrumentation to the lamenting approach to the above line, it's impossible to not be moved by this version of the song.
"All Apologies". The final part with the distorted guitar and Kurt singing "All in all is all we are..." is choke-inducing at least.
The MTV Unplugged in New York album also has tons of this, especially the chilling "Where Did You Sleep Last Night?".
On that note, Kurt Cobain's rare "Do Re Mi" demo, on the WTLO box set. It is one of the last songs recorded before his death, and while a lot folkier and more upbeat than his earlier work, it still manages to be tear inducing.
The acoustic version of "You Know You're Right" is hauntingly stark. Not that the studio version was upbeat or cheery, with lyrics like "things have never been so swell/I have never failed to feel/the pain," but the version that's just Kurt and his acoustic guitar is brutal.
Not to mention when you realize the actual lyrics go "I have never failed to fail". (If the internet is to believed)
"Pennyroyal Tea" can really be depressing (it helps if you pay attention to the lyrics knowing Kurt suffered from gastritis and drug addiction).
"Pennyroyal Tea" is also slang for a home abortion technique.
"Serve The Servants" A song about Cobain's family and childhood. William S. Burroughs, who had collaborated with Cobain, reportedly said that Cobain was already dead upon hearing the song.
"Dumb", with the subtle string arrangements and Kurt denouncing himself for feeling happy. "I think I'm dumb, or maybe just happy... I think I'm just happy..."