I have a lot of songs that strike a sad tone in me, but only a few actually make me feel close to crying:
"Stomach It" by Crywolf, featuring EDEN.
"Breathe Me" by Sia.
"Hymn for the Missing" by Red.
Pretty much the whole album "Hospice" by the Antlers, but especially "Atrophy" and "Wake".
But the one that is always sure to make me cry (at least on the inside) is "Terrible Things" by Mayday Parade. I just can't listen to it.
EDIT:
Almost forgot "Reunion" by Hans Zimmer, soundtrack for Spirit: Stallion of the Cimarron. This song moves me to tears every time, but not only the sad kind of tears. The scene where this is played is one of the best and most emotional scenes in any movie ever, much thanks to this track. It's genius.
Just listen to it for yourself.