This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Banjo-Kazooie
Banjo-Tooie hits you right off the bat by killing off Bottles the mole. It just catches you completely off guard that the sequel to such a bright, fun, cartoony game as Banjo-Kazooie could start out with the death of one of the main characters. And soon afterward you have to face his wife and kids, and Banjo and Kazooie argue for a bit over whether to tell them as everyone goes on about how they expect him home any minute and he's going to be the star of next week's big kickball game.
The state of Banjo's house is just as bad, if not worse. The poor bear and bird's lost just about everything right now; windows are broken, the floor and walls are charred, most of the duo's possessions were destroyed with the exception being a single solitary picture of Tooty dangling from the walls, and all the while a somber rendition of the main theme plays in the background...
The destroyed Grey Jinjo house in Tooie, which has a sign beside it that reads: "In Loving Memory of the Grey Jinjo family, who died quite suddenly and tragically when a giant drill ran over their house."
The death animation. When the duo is killed, Banjo falls to the ground on his stomach, immediately followed by Kazooie popping out of the backpack, dying sprawled on his back.
When you reach the icy side of Hailfire Peaks, you see a cinematic of an alien saucer which is attacked which causes the aliens to fall from the sky and die in the snow. The painful part about this is the dialogue that happens beforehand with the alien saying that he's dead, right before his eyes close.
This turns into Tears of Joy when you use Mumbo Jumbo to revive the aliens and help them fix their ship so they can finally go home.
Clanker. He's part living creature, part machine, his mechanical bits are rusty, Banjo and Kazooie have to shoot him in the teeth, he's chained underwater until the duo go to great lengths to raise him to the surface, he's trapped in a grimy sewer.... this list could go on a bit. And just look at his big, adorable eyes!◊
And that's not to mention how you find him in Banjoland during Nuts and Bolts. He's been dismantled, with one fin acting as a ramp to the snowman, part of his torso acting as a tunnel (complete with rotating blades), his teeth removed and mounted on the wall, and his head hooked up to the museum's heating system. His eyes still blink and follow you, indicating he's clearly still alive and conscious.
Boggy's Igloo in Tooie. There's just something wrong when the normally happy Freezeezy Peak music gets such a halfhearted rendition, all the while you gaze upon such a neglectful slob of a bear, his abusive wife, and their kids who got brattier since the last game.
It got worse in Nuts & Bolts; Boggy has been kicked out of his own house by his wife, and must now earn some notes by running the local gym. Sometimes when you talk to him, he'll be eating his own scarf, commenting on how he hasn't been fed in days.
In Banjo Tooie, you will come across two ice cubes named George & Mildred. Mildred is in the cold side of Hailfire Peaks and has a Jinjo inside of her, and George was blown into Cloud Cuckooland. You must destroy both of them (smash Mildred open and push George into boiling water in Hailfire Peaks) to get the Jinjo and a Jiggy, respectively. It really is unpleasant considering how both of them ask Banjo to help reunite them, and both's final requests are towards their spouse ("Tell George his dinner is in the -" "Tell Mildred I love -")
Thankfully, they both have been resurrected by the time of Nuts&Bolts, possibly by Mumbo or The Lord Of Games.
After the third fight against Klungo in Banjo-Tooie, he comments that every time Banjo and Kazooie beat him in a fight, Grunty beats him and makes his face more disfigured. What really makes it sad is his comment that his wife won't love him anymore if this continues on. It finally makes him quit his job as a minion.
On Spiral Mountain, you find out that amidst the wreckage, a poor little goldfish (presumably from the moat nearby) got trapped under a very large boulder and needs help getting out. Unfortunately, none of your (current) moves from the last game can help you break it, so you must leave the poor fish under the rock until you get a stronger move.
Weldar. Even with his That One Boss status, he's just an absent-minded, near-sighted appliance guarding his Jiggy. And when you beat him, all that's left is his head which still talks. Hard not to feel bad for the guy...