This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / American Dad!
First off, yes there is a Bond parody episode (and a sequel episode to that) called "Tearjerker" where Roger plays a character named Tearjerker, but that's not what we're talking about here, though the movie he produced in the first episode, while deliberately over the top depressing, IS pretty tear jerking in its own right.
This brief moment from "Pulling Double Booty" when Hayley is outlining her three-way with Stan (posing as his body double) and a waitress (though it is more funny than depressing, it does get a little sad when you realize that once upon a time, Stan really did love Hayley and wants her to be the sweet, obedient girl she used to be).
"Weiner of Our Discontent", when Roger realizes that he was sent to Earth as a crash test dummy and not on some important mission and goes through a month-long depression (though it can be argued that Roger deserved it because of his rude behavior — which spanned from the 1950s to the present — before discovering his true intentions of being put on Earth).
Francine facing the facts that Steve is growing up and moving on in "Iced, Iced Babies" (until Steve comes home crying over Debbie breaking up with him and Francine's creepy, "You can stay with Mommy. Forever" line).
"Man in the Moonbounce", where Stan has silly, carefree fun for the first time in his life, then breaks down sobbing. Hell, that whole episode, and the revelation that Stans father leaving meant that Stan had to forego his own childhood to look after his mother, which probably played a big role in turning him into the emotionally stunted jerk he is in adult life.
Stan being Driven to Suicide in "Every Which Way But Lose", saying that he's a big fat loser who doesn't deserve to live because he lost one game in life.
Stan's Best Friend: The story of how young Stan was forced to kill his dog because he thought he had a terminal disease (Turns out it was because his mom was moving into an apartment that didn't allow dogs or cats. Rabbits, however, were only allowed on a case-by-case basis). Even Stan said the story was so sad, that he wasn't going to undercut the flashback with a joke.
The entire episode is a Tear Jerker with Stan being determined to keep Kisses, the family's new puppy, alive after it gets critically injured by a group of pirate cats that crushed it with their hot air balloon's basket. Steve spends several scenes crying his eyes out seeing that his new pet is on the verge of death (which is actually pretty narmy considering the way he screams). The biggest tearjerker comes from Stan slowly accepting the fact that his puppy will never be the same and keeping it alive in its Frankenstein-like state was just making it suffer, so he puts him out of his misery.
Klaus on the brink of tears after Roger rudely tells him that he's not a pet because nobody loves him.
Stan not winning the respect of his neighbors in "I Can't Stan You". While Stan is a Jerkass for most of time, he just wants tobe loved.
The subplot with Lindsay dying from staph infection after breaking her leg in soccer on "Adventures In Hayley-Sitting".
National Treasure 4: Baby Franny: She's Doing Well: The Hole Story puts a depressing new light on all the previous Francine-centric episodes where she either tries to find a career or some new form of fulfillment beyond what she has. Because she was saved from a well when she was younger, and a firefighter died though not really so she could live, Francine's previous attempts at jobs and careers were all to prove that her life really was worth saving. She's felt that she's done nothing with her life, meaning that firefighter died for nothing and it's on her head.
The flashback of Roger's most memorable moments on "Naked to the Limit: One More Time" when Roger is about to go back to his home planet (in the end he doesn't but it still hits home) set to Suicide is Painless (best known as the theme song to M*A*S*H, both the movie and the TV show).
Stan giving Roger the American Flag pin he always wears on his lapel, right before the latter is about to leave.
A large amount of the episode "Lost in Space", but especially the Majestic musical number. Jeff is being shown supposed proof that he never loved Hayley at all, and he cries and falls to his hands and knees at the footage of him not appreciating his wife enough, now with the knowledge that he may never see her again. Scrappy or not, you'll just want to give Jeff a hug after that.
When Jeff confronts the Majestic, he finds out Emperor Zing is forcing it to show only bad memories, or else he'll get killed.
Also take into account of the Fridge Horror, all the alien slaves are being kept from their true love, and probably took the test and failed. One alien is seen crying at the beginning and another was so depressed he hit the bottle (although the latter was also a funny moment), and it's understandable. Also during the start of the revolution, Roger's people and the slaves were all angry at Emperor Zing for his lies to them about love, that they were separated from their loved ones for nothing.
What about the side plot parodying What's Eating Gilbert Grape from "Irregarding Steve" Even in squirrel form, seeing Gilbert and Arnie's mom dying while Arnie desperately tries to make her wake up (thinking she was asleep).
Steve's prom clone Glitter's "death" in "Steve and Snot's Test-Tubular Adventure".
That episodes end credits are played over the image of Snot crying next to Honey's body, holding the macaroni necklace she made him.
In "Office Spaceman", when Steve and Hayley find out about Francine's hatred of left-handed people stemming from her upbringing at a Christian orphanage, they try to talk her into acceptance of lefties. When that fails, they show her a fish just like the one she was repeatedly smacked with to comply with the nuns' teachings. At that moment, Francine breaks down crying in total shame.
Francine:(sobbing) It's true! IT'S ALL TRUE! Oh what you must think of me!
In the "Ricky Spanish" episode, Steve's pet butterfly who, in a jar, had been dropped off the port and into the sea... while still in it's cocoon. How it then exits the cocoon, only find itself trapped until death, is an extremely depressing scene.
The whole of "Independent Movie". The jokes are few and far between, the main story being a genuine attempt by Steve to help Snot come to terms with his estranged dad's death and when the ending hits.
Roger's sad song about his bond with Steve changing and breaking in "E.T. The Abusive Terrestrial". For all the shitty things Roger's done it's hard not to feel bad for the guy.
"The Kidney Stays in the Picture": Stan still going through with donating his kidney to Hayley in "The Kidney Stays in the Picture," despite the chance that Hayley may not be biologically his.
The films that Roger's persona Tearjerker produced. The first film, Oscar Gold, was about a mentally retarded alcoholic Jewish kid whose cancer-ridden puppy dies during the Holocaust. The other? Several hours of a baby chimp trying to revive his dead mother! Considering those films were deliberately designed to get audiences to cry themselves to death...
The scene where Francine accidentally kills a bird that had been 3 days sober. This stuff is to be expected since this is after all, the sister show to Family Guy, but come on. That was just mean.
"The Full Cognitive Redaction of Avery Bullock by the Coward Stan Smith" becomes very melancholic towards the end, when Stan realises that Bullock is suffering from dementia except not really and decides to kill him rather than let the CIA turn him into a vegetable. It winds up becoming deeply depressing watching Bullock having a clear mental breakdown, and watching Stan trying to give the deputy director the best day of his life before going to Mercy Kill him.
Rapture's Delight. Sure, Stan acts like a selfish dick just after the Rapture happens which causes Francine to finally leave him for Jesus, but seven years later, he's a broken man who is blind in one eye and lost his left hand and is forced to fight a battle he doesn't care about for the man who stole his wife. At the end of the episode, he sets an explosion to blow up the building and takes a bullet for Jesus which causes him to slowly bleed out internally and with his last words, he apologizes to Francine for his words before she left him, tells her that she deserves happiness with Jesus. He begs Jesus to escape without him unraptured (and to drag a tearful Francine with him) before dying in the exploding building.
In "Minstrel Krampus", Krampus doesn't punish naughty children because he enjoys it, but because he wants to set them straight, and "We've Been Bad" is about how bad he feels about his job.
Krampus: When I'm breakin' the finger, lord, it breakin' my heart / and every Christmas Eve my soul is always torn apart!
Longest Distant Relationship: Hayley finds out Jeff is going back to Earth and is going through a wormhole to make it there, she agrees to wait. After going through the wormhole, he does arrive 60 years later. She has been waiting for 60 years and everyone in her family is worse off. Later on, she suffers a heart attack and is at the hospital, and Jeff decides to go back and tell Hayley to move on, ending their relationship.
To make things worse, he ended their relationship because he believed that she'd end up with a nice millionaire. The ending makes it seem like in the end, she'll at least be happy with someone again, only for Roger to kill him, leaving her heartbroken and alone again.
Sidney Hoffman's sub plot in "The One That Got Away". Convinced he stole from his credit card, Roger conspires to ruin his entire life, spreading lies to get him fired, dumped by his girlfriend and sabotaged his beautiful garden. It turns out Sidney is in fact a persona of Roger's that took a life of his own to deal with the trauma of his first unselfish thought, Roger ultimately "kills" him after he hires an assassin to stop his conspiracy.
Altough a parody of independent movie cliches, the story in Independent Movie is actually fairly tearjerking on its own.
In "Seizures Suit Stanny", Stan's smartphone lists Jeff as "idiot son-in-law" while Jeff's phone lists Stan as simply "Dad", surrounded by hearts. A major tearjerker considering Jeff's relationship with his own father, and how much Jeff must want to be acknowledged and appreciated by Stan.
In "Holy Shit Jeff's Back!", Haley learning that Jeff is dead (after being dissected by the Collectors AKA the Dissectors), thankfully he gets better at the end.