"Wild Horses". The cover done by The Sundays can also be moving, even if it's mostly because of the Buffy connection. The Flying Burrito Brothers' version, though, is really the one that starts up the waterworks.
"Ran Out of Tears" is another one
"Angie". "Angie... Angie, ain't it good to be alive?"
"Ruby Tuesday". "Goodbye, Ruby Tuesday..."
Not to mention "As Tears Go By" and "No Expectations".
"Gimme Shelter" is dark and moving or outright tear jerker, but listening to the wails of Merry Clayton mixed with the grim lyrics is enough to incite some form of dread in anyone.
While not particularly dark or sad, "Shine a Light" can be very moving, especially the beginning, coming out of the equally moving "Let it Loose" as Exile comes to a close. Something about the gospel air there is to it and that sort of reflective nature of the lyrics makes it stick out.
What happened to Brian Jones before and after he left the band.