"Doubt Comes In" from the musical Hadestown left me sobbing in the theatre when I first saw it performed. I consider it to be the saddest song I have ever heard, because, while I listen to many heartwrenching songs, almost all of the others end with at least a hint of hope.
Honourable Mentions
"Road to Hell (Reprise)" from Hadestown
"Home" from Beetlejuice
"The World Was Wide Enough" from Hamilton
"Cosmic Love" by Florence + the Machine
"Lua" by Bright Eyes
"Breathe" by Kooman & Dimond
(I tried to balance out musical theatre with "normal" songs, so I ended up with a lot)