The last ten minutes of "Where and When". Rebecca wants to share one last kiss with Jack by using H.G. Wells' time machine. When told that the Mental Time Travel is one way and will last nineteen seconds before her time-traveling self turns back into her past self, she replies "Why would I want to come back?"
The ending of "Resonance". When it turns out that the bank robberies were not for the money, but were instead so that an old man could get reunited with his daughter and brought back to sanity again, and when a slow, sad piano song is being played in the background, Pete and Myka can't find it within themselves to report the bank robbers to the FBI, only taking the artifact of the week and leaving.
The third season finale, where Claudia sees Jinx's dead body. That scream will haunt your nightmares.
Also a little before this in the quiet way that Pete, normally the childish one, holds it all in for Claudia and tells her to "Please go back to the car."
Also in the third season finale when H.G. Wells sacrifices herself to create a force field for Pete, Myka and Artie to remain safe in while the warehouse blows up from a bomb the Big Bad planted. The look on Myka's face and the ensuing explosion coupled with the soft piano music of the scene is really something.
In season 2 second last episode, The whole scene when H.G Wells falls victim to the Medusa head, which lets her live out her happiest desire. She is back in Victorian times playing with her daughter. "I'll never leave you again." When she is pulled away, her face is absolutely broken and she then asks Myka "Why did you do that?!"
The first episode of season 4, where Pete, Myka, Claudia, and Artie are traveling around, trying to find the failsafe artifact to fix the warehouse. One by one, they get separated: Claudia gets trapped in a small tunnel, covered in dirt and rocks with iron bars blocking the exit; Myka gets herself arrested to cause a diversion; and Pete gets killed trying to stop someone guarding the final piece of the artifact. The toll it takes on Artie is extremely blatant.
Pete: I won't remember...will I?
Artie: Remember what?
Pete: Being dead...
Artie: No...no, you won't remember.
Artie: ...but I will...
Seeing the Warehouse go up in flames.
It would nearly impossible to name one person who shed a tear when Sykes died the first time around. Then time is reset and Gandhi's cloth removes the evil that infected him when he was a child. He dies, again this time the child that he was asks everyone for forgiveness. Then it dawns on you that he was just another victim of Artifact whammying. Then suddenly, it's not so satisfying anymore.
The episode "No Pain, No Gain".
Claudia: So that's Mr. Frederic?
Mrs. Frederic: Yes. He's my grandson.
Claudia: So your son is...
Mrs. Frederic: Long gone. Not all wonder is endless, Claudia.
A bit uplifting is the fact that, though most her family may be dead, she can still spend quality time with the one she has left. Depressing as it is, Claudia still can't help but smile at the scene.
Same episode, Pete's admission that he would like to have a family someday, but he's resigned to that fact that it may never happen because of his crazy life...unless he settles down with someone who already shares his crazy life (which also may never happen). The last scene of the episode is Pete watching a new father hold his baby, looking extremely sad.
An easily-missed but excruciating one in "The Sky's The Limit".
Steve's reconciliation with his mother over his sister's death 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.
Leena's reaction to Artie losing his marbles, and yelling at the hallucination of Brother Adrian.
Leena had a chance to save herself. She knew Artie was insane and dangerous, Mrs. F had flat-out ordered her to run for it, and she can see auras; she had to have been aware that Artie was no longer himself, and was capable of killing. She stayed because she's Leena, and Artie needed help, and she couldn't leave him.
Evil!Artie in general. He knows exactly what buttons to push and it's just so sad, and they have to deal with this while dealing with Leena's death.
Claudia collapsed against Artie, sobbing because she just had to stab him in the chest in order to save him from himself. Her "I'm sorry" and his "It's okay," signaling that the real Artie had taken back control of his body were heartbreaking.
The antagonist of "3...2...1" is a man who as a child witnessed his father being killed by Joshua's Trumpet (aka the Horn of Jericho), which after a visit by Warehouse Agents, caused him to believe his father had been abducted by aliens and there was a massive Government Conspiracy dedicated to cover it up. Likewise, his several attempts to use the horn to contact his "missing" father end up causing many more deaths as he had inadvertently turned the horn into a superweapon.
Pete's confession to save himself from the effects of an artifact. The story of why he stopped drinking. Because he got behind the wheel and nearly killed his best friend. It's a total heart wrencher.
Steve's death in "Emily Lake". Somewhat undermined by his resurrection later on, but still, at the time that was a real punch in the gut.
Myka telling Pete she has cancer.
Artie's "Breakup Speech" to the Warehouse is gut-wrenching. He's angry, he's frustrated, and he's bitterly, bitterly sad about the possibility of losing the thing he's given his life to for no real reason. Luckily, it turns out that the Warehouse really does care, or at least as much as a building can care about a man.
Pete likewise is having trouble accepting that the Warehouse might move, because he credits the job as making him a better person and is afraid of what might happen if he ever lost it.
"A Faire to Remember": Claudia and her sister Claire getting to spend one final day together. If the decision to make a Heroic Sacrifice doesn't start the waterworks, the sisters' final duet of Garbage's "When I Grow Up" will.