The ending to the episode "The Message," where Tracy finally realizes how much of a Jerk Ass he's been and tries to repent in his last moments, and then his family and the crew are shown gathered around his coffin while he speaks the titular message. The music in this scene is so absolutely powerful.
Also, Mal's choked up comment after shooting Tracy, echoing his words at the beginning of the episode.
"...You've killed me, Sarge." "You killed yourself, son. I was just carrying the bullet a while."
And then that Goddamn little soldier's poem. "When you can't run...you crawl. And when you can't crawl, when you can't do that..." "You find somebody to carry you."
In the special features of the series box set, they actually say (paraphrased) "That music at Tracy's funeral? It wasn't for Tracy as much as it was for the show. We all knew it was going to be over by that point...
On that note, in the fan-made documentary Done the Impossible, Adam Baldwin describes when he was told that the show was canceled. He is visibly trying not to cry. This is during the interview, most of four years later, after the Big Damn Movie. And he still misses the show that much.
Small thing but it grows profound in hindsight - Jayne taking off his "cunning hat" when seeing Tracy's body being handed over to his grieving family grows painful to watch when you factor in his closeness to his own family and his earlier talk to Book about how "[his] kind of life don't last long." It's easy to get "there, but for the grace of God..." vibes from his looks, all because Adam Baldwin is just this good an actor.
Since you mentioned the hat, the look on Jayne's face when the Alliance agent tells him the hat is stupid is gut-wrenching - you know, the hat his mother back home caring for his sick brother made and sent him?
Also, "Objects in Space". Specifically, this exchange:
River: I don't belong....dangerous, like you. Can't be controlled, can't be trusted. Everyone can just go on without me and not have to worry. People could be what they wanted to be....could be with the people they wanted....could live simple, no secrets. Simon: No. River: I'll be fine. I'll be your bounty, Jubal Early, and I'll just fade away.
That's nothing! The looks of guilt on the faces of the rest of the crew really pull on the heart-strings.
There's also River's scene in "War Stories" where she talks about how much she hates being able to think clearly because she knows it will go away, and then breaks down in Simon's arms.
The real power behind that scene is that it establishes just how much of a woobie River is, by showing that she lives in a constant state of quiet despair. The fact that she hates thinking clearly because she knows she can't ever be truly healed just amplifies how painful that scene is to watch.
Really, you can just say "River Tam. Full stop." If she's not being frighteningly creepy, she's breaking your heart with how horribly damaged she is.
"Ariel." "What're you gonna tell the others? ... 'Bout why I'm dead ... Make something up. Don't tell 'em what I did."
Not to mention the brilliant whiplashing effect moments later when Mal seals the airlock, sparing Jayne, and walks away. Jayne, after a few moments of silence, asks if Mal would open the doors to let him in.
Same episode, when Simon scans River and sees what the Feds have done to her. Simon then dumbs it down for Jayne (and us), making it very, very clear how bad it was. On that note, any time River's time at the Academy comes up.
Simon: They opened up her skull and they cut into her brain (...) They did it over, and over.
Adding to that, the scene in "The Train Job" where River wakes up, panics, looks around the ship's infirmary in confusion, gets deeply saddened Puppy-Dog Eyes, and whispers "This isn't home." in this broken little child's voice. It's a tiny thing, but the look of loss and pain on her face is so profound.
The scene in "Safe" where River tries talking to Simon, and breaks down.
River: "I took you away from there. I know you don't think I did, but...I get confused. I remember everything. I remember too much, and...some of it's made up, and some of it can't be quantified, and...there's secrets.... But I understand. You gave up everything you had to find me.....and you found me broken. It's hard for you....you gave up everything you had...." Simon: "Mei-mei. Everything I have is right here."
Made all the worse when River succumbs completely to delusion and tearfully assures Simon that their father (who it's implied was more concerned for his social standing than River) will come for them and that she'll get better.
It becomes better if you think that River was not talking about their actual father, but rather about Mal. (Her new brain patterns use metaphor a lot more than literal fact.)
One really hard-hitting part is when River tries assure Simon that she will get better....but after seeing her breakdown in "War Stories" you realize that River doesn't believe she'll get better. And she's just trying to reassure Simon that she will heal, knowing that she can't, and that she blames herself for the way Simon's life was ruined. It's just so...sad....
(Simon climbs onto the sacrificial pyre and embraces River) Witch-Burning Villager: That's not gonna stop us, Doctor. Simon: Light it.
Let's not forget Mal, Kaylee, Simon and the big sad violin in the pilot, at least the first time you watch it, and until it cuts to that shot of half the crew in fits of laughter... ("That man is psychotic!")
And, of course, the end of "Objects In Space": Early's line, "Well... here I am." Not too moving... until you realize those are absolutely the final words of the show.
"Bullet to the brainpan, squish!" "Don't ever say that."
Anything involving River and Simon is going to be either Tear Jerker or Crowning Moment of Heartwarming or both. Especially "Objects In Space", where River is saying she'll go with Early so everyone can have a normal life, and you can see the look on Simon's face.
Mal settling in to wait for help in "Out of Gas", knowing full well that the odds are stacked against help coming, but willing to try anyway.
In the beginning of the pilot, when Mal realizes that the spaceships he heard are not the the ones he had hoped for the camera zooms in on his face while sad cello music plays. It's not a Tear Jerker the first time around, but after watching the whole series and the movie, you will feel the pain of him falling apart.
Shortly before that you see Mal kiss a cross he keeps around his neck, it really hammers in how much it crushed Mal to lose the war, considering his usual reaction to other peoples faith.
"Jaynestown". There's just something about Jayne managing to convince himself he did something good and worthwhile, then being bluntly reminded that he's still kinda a bad guy. Plus, everything after the mudder's sacrifice is a legitimate Tear Jerker in its own right.
Jayne: All of you! You think someone's just gonna drop money on ya, money they could use? There ain't people like that! There's just people like me.
"It don't make no sense."
During the scene, a sad instrumental version of 'The Hero of Canton' is playing, which on its own would induce tears. It helps if you're a sucker for sad versions of happy songs.
Jayne walking over to scold the downed mudder not yet knowing he's dead and trying to get him on his feet. He's so much like a child in that brief moment.
That tiny little moment in Objects In Space, when Simon, upon hearing what the alliance was trying to do to River, a girl he still thinks of as his goofy, bratty know it all sister, just quietly brokenly says, "She's just a kid, she just wants to be a kid ". It's so easy to forget that he still just sees her as a little girl.
In "Objects in Space" Kaylee managed to take things to a whole new level of sad and scary with one line:
Kaylee: There's... there's nobody can help me.
What makes this even more sad is the fact that Mal said in the pilot episode, "I don't believe there is power in the 'verse that can stop Kaylee from being cheerful." Yet there she was. Crying. A superb Break the Cutie moment.
Rivers entire introduction in the Pilot. Just imagine watching that gut wrenching scene from the actual bay. Up until this point Simon has been cold and aloof, but you can see his heart breaking.
Simon:[about River's coded letter] It just said, "They're hurting us. Get me out."
The funeral scene in Serenity. The fact that even Jayne looks like he's about to break down says something there.
Or when Mal asks Zoe about the repairs near the end. With a look of heartbreaking stoicism, she replies "She's tore up plenty, but she'll fly true."
Also, Mal's final monologue to River. "Love keeps her in the air when she oughta' fall down."
Definitely moving on its own, but if you are willing to take a meta perspective, the monologue gains additional Tear Jerker quality as it seems to be Whedon talking about the show itself getting one final go for the movie.
Book's death scene hit hard. specially the final exchange:
Book: You can't order me around, boy. I'm not one of your crew. Mal: Yes you are.
Book: I don't care what you believe...just believe it.
Every friend, every accomplice, every man, woman and child who ever supported Serenity's crew- Dead. The fact that The Operative, of all people, seems to regret this just hammers in the tragedy.
The Operative:I'm sorry. When your quarry goes to ground, leave no ground to go to.
The Operative's speech to Mal after he organizes the attacks on the crew's allies. The way he enunciates, "I'm a monster," is distressing, to say the least.
River's disjointed breakdown on Miranda, like a lot of the scenes involving River in this movie, was jarring and emotionally hard-hitting, especially the bit at the end where she starts praying frantically in Chinese. In the midst of the chaotic, disjointed Chinese prayers, she suddenly breaks into English long enough to scream.
Please, God, make me stone.
River's ramblings while locked in the supply closet with Simon. It doesn't become apparent just what she's saying for a bit, but then you get her laughing/sobbing comment about "Bullet in the brainpan, squish." and you realize she wants to commit suicide rather than let the rest of the crew be in danger on her account.
Not only that, but when River breaks down before that after Simon hears her mention Miranda.
River: (crying) It isn't mine! The memory, it isn't mine, and I shouldn't have to carry it! Please don't make me sleep again...
River: I don't know what I'm saying. I never know what I'm saying!
Fridge Brilliance when you remember that her conditioning at the Academy was while she was asleep and while she is sleeping she seems to dream often of the Academy and the Reavers. Who blames her for not wanting to be put to sleep.
When Simon gets shot. He's lying there thinking he's about to die and he turns to River and starts apologising that he won't be able to look after her anymore.
"My bag." *THWACK*
When watching the report from the doomed doctor about what happened on Miranda, Wash is the first to realise the truth: "Reavers. They made them." The horror and disbelief on his face is heartbreaking.
Somewhat fridge, but remember when Wash and Zoe joke about her getting a "slinky dress" in "Shindig?" It's what she wears to his funeral.
"I am a leaf on the wind. Watch how I—" Harpoon. Dead. Cue the shocked cries of the fan base.