These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Ear Worm: "Tradition" (especially the fiddle solo), "L'Chaim", "If I Were a Rich Man", and "Matchmaker, Matchmaker" are probably the worst.
Fridge Horror: At the end of the play there's a sad undercurrent, becajuse Hodel and Perchik will apparently remain in deepest Siberia instead of escaping to Krakow like Chava and Fyedka. Then you remember that this takes place about 35 years before the Holocaust, and realize that Chava and her children will likely end up gassed in Auschwitz while Hodel and Perchik may very well survive.
Possibly, but you forget that Siberia was not left untouched by persecution either: in the Civil War, it would be fought over by both the Red Army under Trotsky (who, contrary to popular belief, would casually endorse a pogrom or two once in a while to help "purify" the "counterrevolutionary influences"), and the White army under Kolchak (who, if anything, was even worse in that regard). While it is quite possible that they would escape notice by not being near the railroad, they aren't likely to enjoy themselves either.
Just to complete the picure, if Anatevka existed, there's a good chance it's within the radius of territory rendered uninhabitable by Chernobyl.
Still, thirty years is plenty of time for the others to end up following Tzeitel's parents to America at some point (and hopefully Chava and Fyedka, who go to Krakow, would make the same decision, as Chava and their children would still be considered Jewish under the Nuremberg Laws). Poland had issues with anti-Semitism even before the Nazi invasion, and so perhaps they would have found another reason to pack up and leave.
If they could afford to.
.....You know, let's just draw the line here, and agree that any way you cut it, someone's gonna lose a finger.
"Funny Aneurysm" Moment: the matchmaker's delightful promise that she will make sure their people are fruitful and multiply in Jerusalem. Considering recent events, there's a definite cause to wince.
Not exactly funny, per se, but Perchik participating in the October Revolution definitely counts.
Germans Love David Hasselhoff: The play was a surprise hit in Japan, where the theme of crumbling tradition resonated heavily with the elderly generation.
Nightmare Fuel: The movie version of Fruma Sarah's solo in "The Dream".
One-Scene Wonder: Fruma Sarah. It's also a theatre role where hamming it up is required, along with a harness (or a cast member with a sturdy set of shoulders).
Protagonist Title Fallacy: Teyve is not the fiddler on the roof; however, the fiddler represents the inhabitants of Anatevka: trying to play a pleasant old tune in perilous circumstances.
Values Dissonance: The original Tevye the Dairyman stories by Sholem Aleykhem, on which the musical is based, portrayed Tevye's decision to disown Chava after she marries Fyedka in a more positive light. It was what any good Jewish father of the time would do if his child chose to marry a Gentile (and to really hammer this home, Aleykhem has Chava abandon Fyedka and return to Judaism at the end). People today who see the musical (which doesn't really take a side), though, usually come away from it thinking Tevye's actions toward her are excessive and cruel.
Though the movie does show that the father feels trapped by his traditions at this point, but also that she was kinda stupid to run off and expect everything to be hunky dory. Still, it is easy to see it as a Kick the Dog moment for Tevye.
And it's important to remember that it's not just culture, it's religion. By accepting Chava's marriage he would be rejecting his faith, and he wasn't willing to do that. In fact, to him his daughter rejecting their faith to marry a Christian is worse than her dying, as he believes it has eternal consequences.
He does unbend a little bit at the very end when the family is having to leave their land and Chava comes to say goodbye. Specifically, he gives her a quiet "God be with you" and subtly endorses his wife and other daughter's reaching out to her.