One of the first in the movie is when Belle is crying because she'll never be free to leave the castle or see her father again.
Belle (weeping): I never got to say goodbye.
The look on Beast's face shows he truly feels remorseful for the pain she's feeling, and this is still when he was particularly savage and cruel. He then offers her a room instead of the dungeon, and gives the same expression when he notices a tear going down her cheek.
This bit after "Gaston"? Maurice is trying to persuade the others that Belle, who happens to be the most beautiful girl in town if nothing else to them, is in grave danger (well, as far as he knows):
Maurice: Will no one help me?!
Even in the midst of his Nightmare Fuel-saturated outburst after discovering Belle in the West Wing, you can hear a hint of anguish in the Beast's voice as he roars at her to get out. He's not just angry that she disobeyed his orders, but was terrified that she could have damaged the rose and left him trapped as a Beast forever. And after she runs off, his angry expression slowly turns to horror, and he covers his face with his paw in agonized remorse.
The people who animated the Beast are sick bastards. You can feel his heart break when he tells Belle to leave. Following her exit:
Beast: ... I let her go. Cogsworth: (chuckles) Yes, yes. Splen— you...what? How could you do that?! Beast:I had to. Cogsworth: Yes, but... why? Beast:Because... I love her.
And then when he's on the balcony and roars while watching Belle ride away from the castle. The musical version makes this scene even more tearjerking: after Cogsworth, Mrs. Potts, and Lumiere walk away, the Beast sings a reprise of 'If I Can't Love Her.' One version of the song:
So I finally know That I shall always be In this hopeless state And condemned to wait Wait for death to set me free
To make matters worse, the despairing Beast is what the villagers are shown when Belle uses the mirror to prove he's real (as the mirror shows exactly what's happening at the moment its magic is invoked). All they see is a ranting monster (he is, in fact, weeping), and despite her best efforts, Gaston plays on their fear to convince them to kill him.
The actual scene where the Beast releases her is completely heartbreaking because it meant two very different things to the characters. Belle has no idea just how badly the Beast is hurting, nor does she realize the sacrifice he's making for her.
Belle: Papa! Oh, no, he's sick, he may be dying! And he's all alone! Beast: (looks at the rose and realizes what must be done) Then... then you must go to him. Belle: What did you say? Beast: I release you. You are no longer my prisoner. Belle: You mean... I'm free? Beast: Yes. Belle: Oh, thank you! Hold on, Papa, I'm on my way. (she turns to leave only to turn back and hand him the Magic Mirror) Beast: Take it with you, so you'll always have a way to look back, and remember me. Belle: Thank you for understanding how much he needs me.
Later, Gaston readies an arrow to fire at the Beast. The Beast looks at him, but just turns away, too dejected to care about his own safety. Even when the arrow strikes him, he's too depressed to fight back. This is the true meaning of Despair Event Horizon.
And he doesn't even care that he's mocking his kindness. "What's a matter, Beast? Too kind and gentle to fight back?!" He just looks at Gaston as if to say "Just end it. Please."
After being stabbed in the back by Gaston, the Beast is lying in Belle's arms (the first person to make him feel anything other than self-hatred and loathing in years), bleeding, obviously dying, struggling to breathe, let alone talk.
Beast: You came back... Belle: Of course I came back. I couldn't let them... oh, this is all my fault! If only I'd gotten here sooner... Beast: Maybe... it's better... it's better this way... Belle: Don't talk like that. You'll be all right. We're together now; everything's going to be fine. You'll see. Beast: (he reaches up and touches her cheek) At least... I got to see you... one last time... (his eyes close)
And the tears keep coming when we see Belle's response... *sob*
Belle: No... no! No, please... Please don't leave me... (she sobs softly, laying her face against his chest) ... I love you...
It's a testament to the skill of Paige O'Hara and Robby Benson that the dialogue reduced the filmmakers themselves to tears in that scene while they were recording it.
When the Beast lets Belle go to find her father, it's painfully clear he doesn't expect her to come back, doesn't think she would ever want to come back. So, just his expression later, even though he's dying, when he says, "You came back..." *bawl*
After Belle leaves, Mrs. Potts tries to warn the Beast of Gaston and his mob's approach.
Mrs. Potts: But Sir... The castle is under attack! Beast: It doesn't matter now. Just let them come.
After that, the In Memoriam to Howard Ashman, the lyricist for the film who worked with Alan Menken for several other films of the Disney Renaissance, and who died before it was released.
"To our friend Howard, who gave a mermaid her voice and a beast his soul, we will be forever grateful."
A very small and brief one occurs with Lefou of all characters. After Gaston throws him in the snow and tells him "Don't move from that spot until Belle and her father get home!", you can feel the pain in the little guy's voice as he says "But... but..." and watches his buddy leave. It only lasts a second, but Lefou looked like a kickedpuppy for that brief amount of time.
There's also a little Fridge Horror involving LeFou. How DOES he react to Gaston's death anyway? Of course he could have pulled a Heel–Face Turn, finally seen Gaston as the true monster he is and happily moved on, as he explicitly does in the live-action remake. But if not (and we have no way of knowing), then imagine how devastated he must have been to lose his idol in such a horrible way!
The Beast's songs in the musical are Tearjerkers, especially "If I Can't Love Her." It's not bad enough that in the movie, you feel his heartbreak and despair—in the musical, he sings about them. It really makes the Beast sympathetic.
No pain could be deeper.
No life could be cheaper.
No point anymore if I can't love her.
No spirit could win me,
No hope left within me.
Hope I could've loved her and that she'd set me free.
But it's not to be
If I can't love her...
Let the world be done with me!
After "Something There" in the musical, this dialogue occurs:
Chip: Mama? Mrs. Potts: Yes, Chip? Chip: Will I ever get to be... a boy again? Mrs. Potts: ... I hope so. Chip: When will I know? Mrs. Potts: ... Soon. If it's to be... it will be very soon now.
As mentioned on a few other pages, the musical, rather than simply having the servants be turned into walking, talking household knickknacks, instead sees them slowly and irreversibly transforming into inanimate objects (which will either kill them or leave them as humans stuck in unmoving, unfeeling forms forever). Here, then, Mrs. Potts is forced to admit to her only son that Belle may not break the curse in time. She's essentially a mother telling her child that he, herself, and all of his friends may be going to die, and they're completely powerless to stop it, as they can't force the Beast to genuinely love someone. It's a quiet moment, but it stings.
While the scene of Cogsworth of too embarrassed to show the fact he now has a windup key on his back and backs away so nobody can see it in the stage show is Played for Laughs, it's a sobering reminder of what's coming their way.
Similarly one of their discussed coworkers who is described as "dumb as a brick" was turned into the brick wall behind the stove. At least Cogsworth, Lumiere and the others like them can move and talk...
Prior to that is his line "She'll never see me as anything but a monster." with the understanding that this is how the Beast sees himself.
The Beast's first solo in the musical "How Long Must This Go On?", is the first time we see just how much the temperamental Beast hates what he's become, recognizes that he messed up but feels that it's too late and a painful reminder of the human he once was.
Beast: How long must this go on, this cruel trick of fate?
And then that witch was gone, and left me in this state,
An object of revulsion and derision.
Is there no one who can show me how to win the world's forgiveness?
The transformation scene has one tearjerker tucked in with all the happiness and triumph: Look at Beast's eyes or more specifically, at the very noticeable bags under his eyes, even when he's got Mrs. Potts, Lumiere, and Cogsworth swept up in a big group hug. You can't write this off as Beast simply being exhausted from being dead not two seconds before. The bags are still there in the ballroom scene that follows. It's a very subtle and heartbreaking way to tell the viewer he's been through hell and back and is very likely still carrying a great many mental scars from it.
The Prologue that shows how the Prince was cursed to become the Beast can invoke tears as well, because you do see why he's isolated from the rest of the world and how that could make him lose hope that he'd ever be free.
Narrator: As the years passed, he fell into despair and lost all hope. For who could ever learn to love a beast?
It's pretty disheartening to see Lumiere, who has been described as a lighthearted character who almost always has a smile on his face and holds onto hope the longest the curse would be broken, wonder if it might have been better if Belle had never shown up after the Beast lets her go.
Also, Cogsworth, who was the most skeptical one from the start, was glad to see his master happy and was eager for things to return to normal. Until Beast lets Belle leave. Later he complains to the others that he knew they shouldn't have gotten their hopes up.
The first trailer for the 2017 film has a very melancholy feel, showing the nearly-empty castle, having clearly fallen apart over the years. It does, however, end on a hopeful note with Belle showing up. It can also induce happy tears for fans of the original, the image of Belle finding the enchanted rose being particularly striking.
During "Something There" Belle has enticed a bird into Beast's hands. She wanders off as Beast smiles behind her. Then his face falls and he looks away, conveying with body language the message "I was just fooling myself, she could never care for me."
After Belle runs out of the castle after the Beast frightens her for entering the west wing, Lumiere and Cogsworth try to stop her to no avail. It's especially sad to hear Cogsworth's quiet, desperate, "No, wait, please wait." Though he's treated as a Butt Monkey, he's been the one most resistant to letting Belle stay - because he feared this would happen. He wants to be human just as much as everyone else, but his pessimistic nature means he's never really believed it would happen. Belle fleeing just affirms what he's been thinking from the start, that there is no hope.
In the musical version, when the Beast discovers Belle with the rose, he scares her out of the room. But then he follows her, furious but trying to keep her from running away. He tries to grab her arm (not to hurt her, just to stop her from running), only to accidentally tear the sleeve of her dress. There's a moment of shocked silence, and just as the Beast says "I'm sorry", Belle has fled in terror. The Beast is left behind, calling out "I'm sorry! I didn't mean to frighten you!"