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 genuinely 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 Mankey flipping out and attempting to destroy your base.
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 detailing 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 Kecleon 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, a letter from Pelipper suggests that one of the Kecleon who sells 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?
Moltres: No. I was only preparing to take off.
During your rescue mission to save Shiftry in Silent Chasm, when your partner hears the story of a monster (actually Zapdos) residing in the depths of the dungeon, they fake a stomachache to get out of it. Either the player character can jump in on the act, or misread the cues entirely. Both options end the same, as faking it results in Jumpluff pointing out that you haven't eaten breakfast yet, while saying you're fine causes the partner to get angry at you for missing the cues.
Becomes a Brick Joke later on, as Team A.C.T. tell you about how dangerous Zapdos is, saying it's very powerful, it's electic attacks will leave you stunned and shaking, and that it might even try to eat you. The player character can say they're not scared... or they can feign a stomachache. If you do so, the partner says that you do this act all the time, and that it's a real headache.