Coldplay/Tear Jerker

"Fix You" is a major one, and it's a good song for dedicating to one's offspring "who had a tough year"...
"Warning Sign" from A Rush of Blood to the Head is another one.
As well as "The Scientist".
"'Till Kingdom Come" can also make one tear up... partially because it was written for Johnny Cash to sing, but he died before he got the chance to record it.
Then there is "Rush of Blood to the Head". It can remind one of the moment in Forrest Gump when Forrest mentions how he paid for Jenny's childhood home to be demolished.
"Amsterdam", from A Rush of Blood to the Head. Holy crap.
Be my mirror, my sword and shield, my missionaries in a foreign field/For some reason I can't explain, I know St. Peter won't call my name/Never an honest word: that was when I ruled the world.
Christmas Lights, especially during the Christmas season.
"Strawberry Swing".
"Violet Hill" might be another one. Although, it might also be a good song to play when you really really want to hit someone -- but you know it's not a good idea.
"Clocks", to name yet another.
"Don't Panic," as well.
"Trouble". Just try listening to it. Just try.
"See You Soon" is heartbreaking.
"Death And All His Friends".
"Sparks", but maybe for the reason that it sounds comforting to cry to, rather than provoking any tears.
"We Never Change".
New song "Us Against The World". Parachutes-style acoustics + desperate, yet hopeful lyrics + beautiful vocal harmonies near the end = Manly Tears.
"Moving To Mars". "We won't see the earth again..."
"Shiver" is pretty depressing.
"Yellow" can range from being a pretty depressing song to possibly being the happiest song you've ever heard, depending on who you are and how you interpret it.
"Lovers in Japan". Every time.
"Up in Flames" is devastating.
Easy to Please, as mentioned on the Crowning Music of Awesome page.
"Paradise" in the heartwarming sense...
Their tribute to the Beastie Boys. The first time they covered "(You Gotta) Fight For Your Right (to Party)" was a light-hearted get-well-soon tribute, but when they covered it after Adam "MCA" Yauch's death, they managed to turn it into a heartbreaking performance.

I said Mom you're just jealous, it's the Beastie Boys
I said Mom you're just jealous, it's the Beastie Boys
And so we're sending all our love to the Beastie Boys

The warm, soothing beginning of "Life in Technicolor".