The way that your partner reacts if you choose any rude dialogue options. (Such as disagreeing to joining a rescue team, agreeing that they're a wimp, etc.) The fact that they're geniunely shocked that you wouldn't be as kind as humanly possible can be hilarious if you think about it enough.
Moments for the Red/Blue games
Hey, don't forget about the first game! Just read the beginning of this page.
From the first game, the conclusion of the Rescue Team Base renovation. In order to get the Mankey Gang to assist in renovating, you have to give them chestnuts so they're actually motivated to help. And then once the base is finished, they realize that you won't give them any more chestnuts. Cue the Mankeys flipping out and attempting to destroy your base.
In either of the sequels, any enemy with Pluck (usually Vullaby near the beginning of Gates To Infinity) can take and eat a random berry or seed from your inventory. Frustrating when it gets your healing berries, or worse, your Reviver Seeds. But sometimes it'll grab a Totter or Sleep Seed...
This part after you and your partner rescue Diglett from Skarmory:
Diglett: My feet feel like they're still walking on air...
(The two Magnemite, you, and your partner sweat drop)
Your Partner: (Feet...?)
Magnemite: (HE HAS THEM? FEET?)
In Red/Blue Mystery Dungeon, you get letters from Pelipper detaling jobs and how to play the game. Post-game the letters discuss Pokemon mysteries, like what Shellder and Shelgon look like inside their shells. This world is entirely habituated by Pokemon and not even they know what they look like!
In every Mystery Dungeon game, Kecleon only appears as a merchant to sell stuff to you, and in dungeons if you steal from it an endless horde of IMMENSELY high leveled Kecleons will warp in and defeat you, unless you are as high leveled as them, and even then you're still pretty much doomed. In Red/Blue Mystery Dungeon, the first one, a letter from Pelliper suggests that one of the Kecleon who sell to you in the square just ventures out into dungeons to sell stuff and gets angry easily.
During the scene where Gengar is telling the villagers about the player supposedly being the human from the Ninetales Curse, Wigglytuff is at the side, behind the others, moving up and down while trying to get a view of what's going on.
Before Moltres leaves after you defeat him, after scaring you twice:
Your Partner: Waaaah! Is...is there anything else?