In the pilot, when Pete tells Myka about his tragic past and why he now always follows his instincts. Also counts as a Tear Jerker.
In "Don't Hate the Player", everyone from the Warehouse has been programmed into normally-humorous parodies of themselves. Leena's character... is an angel. A literal angel.
When Rebbeca finds the ring Jack was planing on giving her at the end of "Burnout".
Also, the fact that both Jack and Pete were strong enough to resist the Artifact's influence and tried to protect others around them by sealing/destroying it even at the cost of their own lives.
When Pete and Myka were reunited with resurrected Artie in "Time Will Tell".
When Pete told a nervous Myka that she really was beautiful in "Age Before Beauty".
Myka's resignation letter to everyone in "Reset. Also a Tear Jerker.
"Secret Santa". A family is brought back together, Artie reunites with his dad, Claudia makes t-shirts for everyone (based on artifacts that nearly killed them) and the extended Warehouse family have dinner together.
In "Trials," an old man steals Typhoid Mary's knife, and Claudia and Jinks pursue him thinking he will misuse it. In actuality, the man's son is suffering from leukemia and will soon die - leaving behind a wife and child. However, the man uses the artifact to transfer the leukemia to himself - sacrificing his own life so that his son can have a full one.
"It wasn't fair. Now it is."
When Artie uses some Artifact glasses to help the young boy Danny's father in "Insatiable".
Later when Danny noticed Myka was sick like his father and offered her the glasses.
Pete hugging HG Wells. Given how much he didn't trust her, it's quite heartwarming.
In hindsight, one for the warehouse itself. We know that the warehouse is a sort of quasi-sentient thing insofar as it can like people. Thus when HG sacrifices herself to save the team and she smells apples which is the warehouse's way of saying it likes someone, it means that the warehouse knowing it was going to be destroyed was telling HG that she had been redeemed in its eyes for saving the team which, by proxy, it also liked. It also means that the warehouse was telling HG that everything was okay - that as long as the team was okay, it was okay with her choice and with its own impending death.
Pete does it again when he gives Mrs. Frederic a hug at the end of "The Greatest Gift", even if everyone (including Pete and Mrs. Frederic) all felt a little awkward about the whole thing.
In the season 4 premiere, Pete's dying moments, as well as Artie's heartbreaking reaction. "I'm not gonna remember... dying?" "No... you won't remember. But I will... I will."
In "An Evil Within", the entire scene where Claudia revives Steve with the metronome. Particularly when she cries after he regains consciousness.
"No Pain, "No Gain", where Pete acts as Myka's "sassy best friend" throughout the episode. Especially after they learn that the dogtag artifact only works on someone that the user loves. Would be more heartwarming if Myka responded in kind, but still.
Artie wishing Vanessa and Hugo all the best when he thinks she is in love with him and Claudia's super excited reaction to them having their first kiss.
Artie: Uh, listen, I know that we had dinner plans, but I hope that you know I wish you and Hugo nothing but the best. Well, certainly, if I'm wishing him the best, I'm wishing him you, because...
Steve's reconciliation with his mother over his sister's death in "Second Chance" is both a full-on Tear Jerker and Crowning Moment of Heartwarming. That Claudia's expressions and mannerisms constantly reminds them of the dearly departed is just the icing on the heartbreak cake.
Surprisingly, in "We All Fall Down", the events leading to Claudia stabbing Artie with the dagger. Claudia's reaction to Artie's words, bringing her to tears, and afterwards, when she openly cries in Artie's arms, really brings it home.
Claudia's utter refusal to think of Artie as the one who killed Leena. She's the only one who really, truly differentiates between evil!Artie and the real Artie, and furiously corrects anyone who suggests otherwise. Considering that Claudia was the one who suffered the most from evil!Artie's cruelty, her love for the only father figure she has is highlighted more now than ever.
Myka: I mean, he... *quieter* He killed Leena...
Claudia:No he didn't! That wasn't Artie!
Artie creates a haven for himself, inside his mind. In his world of perfect peace and safety, it's just him playing the piano softly at Leena's, with Leena alive and happy and making cookies just like always. Crosses over with Tearjerker.
Claudia and Jinks rescuing Artie from the coma he put himself in—because even if he thinks the world is better off without him, his Agents disagree.
Jinks's pep talk to Claudia after she has just had enough of the Warehouse and what it has cost her.
Claudia: I hate this. The Warehouse was supposed to do good and instead it just keeps killing the people I love.
Jinks: Well, that's kind of a simplification.
Claudia: Is it though? You died. Leena died. And now the whole damn world is dying.
Jinks: Claud, Claud, Claud. Pete and Myka will find a cure. They will.
Claudia: So what? It's not going to save Leena. And Artie? I stabbed Artie with a dagger. Why bother if we just keep losing?
Jinks: Because it's not the win, it's the fight. Evil is relentless. It always has been. You take one evil out and another pops up to replace it.
Claudia: Worst. Pep. Talk. Ever.
Jinks: But if we stop fighting, if we give up, what does that say about us? I knew that Sykes was dangerous but we had to stop him. Mrs. Frederick warned Leena to get out but she stayed to help Artie. And you? You knew the world needed saving even if it cost you the only father you've ever known. The fight. That's what matters.
Artie agreeing to see a therapist to deal with his guilt over killing Leena after she makes an impassioned speech about how this guilt will consume him.
Pete and Myka giving Monty the Amazing a tour of the Warehouses various magician related artifacts.
Claudia's speech the end of "Instinct," where she reveals that, despite everything that's happened, she really actually does want to be the Warehouse's Caretaker, she thinks of the Warehouse as a living thing that's upset with her for "stabbing its husband", and wants to make it right. And then, at the end of the episode: "That's funny...I thought I smelled apples."
The discussion that Steve has with Abigail in "What Matters Most" where Steve expresses his concerns about never finding that "one special person" to share his Warehouse life with. Abigail tells him that, actually, he has four special people that he can share all of his life with, and maybe he should try to open up to them more. They wind up both taking her advice, with Abigail sharing her photography hobby with the others, and Steve cooking dinner for everyone.
Claudia and her sister Claire singing together in "A Faire to Remember."
In the series finale, it seems like the Warehouse is going to move, and Artie rants about having given his life to protect it without so much as a thank you. A gust comes out of nowhere and an apple rolls down the aisle to his feet. The Warehouse didn't just make him smell apples like it normally does to show affection, it actually gave him a whole apple. That's how much it cared for him.
Artie: You're welcome.
Artie with another one. After revealing to the team that he has a son who he's had to keep secret to protect him from the enemies of the Warehouse, Claudia is particularly hurt, because of the father figure relationship that she has with Artie now feeling dishonest. (Plus, considering what happened to Claire and Joshua, the idea of her having someone who'd effectively be a sibling figure that she's never met and likely never will meet is going to hurt.) They go through a classic father-daughter conversation, with Artie telling her why he's never told her about him, that ends with Artie telling Claudia that he's come to most appreciate his relationship with his son because it makes him a better father for her.
Pete's speech after he reveals that he has no defining moment because they're all defining moments
The smile on Future! Claudia's face when the Future! Warehouse team reminds her of Myka, Pete, and Artie