A lot of songs from Fiddler on the Roof tug at the heartstings. Probably one of the most infamous examples is "Sunrise, Sunset". And when done right, "Sabbath Prayer" also has a very powerful feeling to it-you'll cry and you won't know why. "Far From the Home I Love" is also heartbreaking, as well as the lines immediately after:
HODEL: God only knows when we shall see each other again.
TEVYE: Then we shall leave it in His hands.
And the song Tevye sings about Chava, and his heartbreak, and her heartbreak.
When Tevye disowns Chava and the song he sings about it afterward.
Anatevka. The song is just so bleak. They try to kid themselves into saying the village was nothing, just a pile of rubbish, it should have been destroyed ages ago. But they know this is a lie. It wasn't much, but it was their home dammit!
Then there's the scene when the rabbi is preparing to leave, with the scrolls and things, he looks at the synagogue, one that has been around for generations, with writings and teachings on its walls, and he has to leave. What makes it so bleak and tragic is the silence and the expression on the old man's face and his praying under his breath.
"Sun rise, sun set", it's a beautifully somber song about watching your children grow up and leave you, and how it's a wonderful thing to watch, but doesn't make the fact that you are growing older and left alone any easier. For obvious reasons, it is a very popular choice for father-daugther/mother-son dances at weddings (at least Jewish weddings).
"L'Chaim" can count as this. Here we have a song that involves the Jews and Gentiles putting aside their personal biases to celebrate the announcement of a wedding. It all feels worthless during the wedding when the reception is destroyed and the Jews terrorized.
The look on the constable's face when the inspector leaves ordering him to do a sort of pogram. Its almost a beaten, self-loathing expression. He clearly doesn't want to harm the Jewish citizens but he has and compromises and he doesn't like it for one minute.
The pogrom that occurs at Tzeitel and Motel's wedding. Previously, the constable and the other Cossacks was celebrating with Teyve over his daughter's marriage. A happy event suddenly ends with all the guests running in terror, the whole reception wrecked, Tevye and his family left to clean up the mess, and the constable can only weakly mutter, "Orders are orders."
Tevye's casting out of Chava. Her heart is breaking, his heart is breaking, and if yours doesn't too you don't have one.