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)