"Make This Go On Forever", especially with his pleading "Please just save me from this darkness..."
"Lifeboats" can make one feel calm and sad.
"Run". That is all. It can be especially affecting for people who have lost loved ones.
"The Finish Line". Although the lyrics are a bit ambiguous — it might seem like it's talking about someone who's been in a car crash, and dying. It really doesn't help that at the end, the heartbeat-like sound that had been going on all through the song, stops. However, it's ambiguous enough to be about everything ending in general. And that's the sad part. He says it himself:
"It's like the finish line...where everything just ends..."
"You Could Be Happy" is another one. The music box melody at the beginning can make some people tear up, while the sheer stoicism of the lyrics can hit then even harder. It's about a guy who, after a break-up, still loves his girlfriend and misses her. But he wants his beloved to be happy.
"Chasing Cars", especially during the last chorus. The music has almost faded, you think it's over, and BAM. The emotion there is overwhelming.
It's also worth mentioning that people even tend to use this song in music videos they create that are meant to be sad. For example, the already heartbreaking anime film Grave of the Fireflies has a video tributed to it with this song used as accompaniment, and the song actually makes the montages from it even sadder for some people than the film itself! Now that's a big accomplishment.
Off the same album: "Set The Fire To The Third Bar". A duet with Martha Wainwright about loneliness and separation, set to fitting music. Beautiful and heartbreaking at the same time.
Somewhere a Clock is Ticking. A song about death, and as the song goes on the deaths get more obvious and complex. At the end the song reminds us that:
"A clock is ticking, but it's hidden far away. Safe and sound... Safe and sound..."
The third part of the last track on their 'A Hundred Million Suns' album, 'Daybreak'. Add to this the gorgeous heartstring-tugging melody, and you might end up sobbing.
"And in the middle of the flood, I felt my worth When you held onto me like I was your little liferaft.\
Please know that you were mine as well."
Also, 'Grazed Knees'. Whatever your interpretation of the relationship portrayed in this song, it's crushingly sad.
"If There's a Rocket, Tie Me to It" Bonus points if it is a version where Gary Lightbody looks like he's about to bawl his eyes out.
"It's Beginning to Get to Me"
The unrequited, longing sadness of "One Night is Not Enough" & "An Olive Grove Facing the Sea"
A B-Side to "Chasing Cars" is a song called "It Doesn't Matter Where, Just Drive" It is about a miscarriage, and the loss of never getting to see that child's life played out.
The video to "This Isn't Everything You Are". The lyrics are meant to be a tearjerker-antidote, but Gary Lightbody is clearly holding back tears throughout the session. The most poignant part of the whole video is the moment when he chokes up, stops syncing to the chorus, and stares straight into the camera.
Called Out In The Dark follows suit with its Tear Jerker tendencies. The lyrics and video press how people are constantly trying to find somewhere they belong and aspiring to their dreams.
"My Brothers" off the Called out in the dark e.p is a tearjerker, especially with Lightbody's pleading "go easy on me, my brothers..."
"The President", a beautiful, absolutely heartwrenching song about failure.
"New York" is a poignant song about heart-break and having your significant other, the person you'd considered to be your soul mate, move very far away (or as the song implies, to New York).