The cutscene after you accomplish the final mission in Star Fox 64 after playing the HARD route. The final boss had been defeated, but the complex was exploding with Fox still inside... Then his father appears and leads him to safety before vanishing. To underscore it, if Peppy was not shot down in the battle with Star Wolf, when Fox is looking around trying to figure out where his dad went, Peppy asks Fox if there's a problem. Fox stammers, "N-nothing. Nothing's wrong." but with the original Japanese audio, you can tell Fox is fighting back tears.
From Star Fox Assault: Peppy and Rob's Heroic Sacrifice. Particularly the part where Fox says: "Peppy... you mustn't."
General Pepper's fate during the Corneria stage in Assault. The guy spends the entire game doing his best to guide Star Fox so they can stop the aparoids. Sometime when they were off saving Sauria, Corneria is invaded and Pepper's forced to watch most of the planet's defenses be slaughtered. As he's in the middle of calling to Star Fox for help, he loses the connection and presumably is captured by the enemy. And at the end of the stage? We see that the aparoids have infected his flagship... while he's still inside. He spends the entire battle shouting for Star Fox to kill him so he won't be fully assimilated and forced to work for the aparoids. He survives, but still!
Say what you will about it, but the ending melody to the nine endings of Star Fox Command is heart-bending and tear-worthy, especially considering that most of the endings are very bittersweet or outright depressing. The humble melody invokes a feeling that someone is saying good-bye for the last time, or that hope is fading...
Or, if you get the "Good-bye Fox" ending where Fox and Krystal retire, marry, and have a baby, the tearjerking melody actually feels happy.
Star Fox 64 3D has a theme the original didn't have (Credits Theme 2), and it makes you feel so good if you played the original first and then this version.
Blink and you'll miss it, but in Star Fox 64, Falco's reaction ("I can't believe they did this...") to Zoness' pollution by Andross' experiments seems a bit more personal than the rest of the team. Considering how out-of-character it is for his usual cocky attitude, it makes you wonder if he knew what the planet was like before it was ruined...
For most of 64, if Fox is killed in battle, his wingmen scream for his life. However, during the final sections of either Venom path, when he's taking on Andross alone, if he dies, you hear his screams instead. It's a pretty heartwrenching incentive to stay alive.