Louise admitting that she only wanted to spend time with Bob and Gene again in "Spaghetti Western and Meatballs".
In "Bob Day Afternoon", a bank robber across the street takes in hostages and orders burgers from the restaurant as one of his demands. When the robber wants Bob to deliver the burgers himself, Linda and the kids cling to him and plead with him not to go.
Teddy having to deal with his guinea pig (accidentally) getting crushed to death by Bob in "Full Bars". Doesn't help that he's so emotionally distraught over it that he stops his Halloween party and doesn't let anyone out until he finds out who killed her.
Bob's flashback of his shitty childhood in "Bob Fires The Kids".
This line from Linda to Louise in "Mother Daughter Laser Razor":
Linda: I just wish you liked me, is all.
The Belcher family freaking out in "Topsy" when it looks like Tina was electrocuted by Louise's project. Bob and Linda rush to Tina's side, hysterically calling her name while Bob cradles her body, Louise is horrified that she may have accidentally killed her sister, and Gene looks and sounds like he's on the verge of tears. It's hard to blame them for being so relieved when it turns out Tina was just acting.
Louise breaking down and admitting that she's afraid to get her cavity filled in "The Kids Run Away" is pretty heartbreaking.
During the "Bad Things are Bad" song in the season four finale, Bob is lamenting on how he's going to die alone by drowning under a pier. While the rest of the song and pretty much the entire series is comprised of genuinely saddening moments, that in particular is pretty darn depressing.
Tina's story of what would happen if Bob met Linda and he didn't have a mustache in, "Sliding Bobs." Hugo marries Linda, has backwards versions of the Belcher kids, and Bob becomes a health inspector.
Louise's moment of clarity and self-loathing in "Nice-capades" when she realizes if she was actually a nice person, she wouldn't need to put on a show about it.
Rudy's breakdown at the end of "House of 1000 Bounces" after the kids are detained in the ranger's station. Rudy starts screaming at Louise that he didn't want to steal the bounce house but no one would listen to him, and while his idea for a "spoon puppet show" seemed like a pathetic way for him to salvage how crappy his birthday was going it turned out he'd written a fairly detailed script and wanted to perform it with the other kids. He starts crying over how horrible his birthday has gone, but thankfully Louise steps in and tells Rudy they can still put on the show he wanted.
Tina having to say goodbye to her imaginary horse Jericho in "The Horse Rider-er".
Bob's rant near the end of "Glued, Where's My Bob?", where he says that getting stuck to the toilet and publicly humiliated is just the sort of nonsense he has to put up with every day at Bob's Burgers. You can tell the poor guy is at the end of his rope.
"Large Brother, Where Fart Thou?" has Louise being driven to tears when Logan finally corners her.
"Thanks-Hoarding" sheds some light on Teddy as a child; he developed a hoarding obsession due to how he's always felt that he alone could fix anything, even his parent's failing marriage. By the end, you just want to give him a hug and tell him he'll be okay.