Bright Star (although it turns out happy, but man the end of Act 1 got me both times I saw it as did several scenes in Act 2). Miss Saigon, Next To Normal.
I found The Life to be sad. Memphis' brutality and Fleetwood's many flaws are sad enough, but the lives of the pros are pretty dire. Sonya's excellent "The Oldest Profession" places a spotlight on the struggles of 'the life,' especially as an aging hooker. Queen's songs "We had a Dream' and "He's no Good" are heartbreaking, especially the latter song (it hit home for me). I was a mess at the end of Act II when Queen and Sonya sing "My Friend" and Sonya sacrifices her freedom so that Queen has an opportunity for a new life. I still cry when I hear a recording of those songs and I'm transported back to those moments in the theater.
Although Chess was sad, as in a hot mess, the song "Pity the Child" is heartbreaking.