Several of the phone calls you can get in HG/SS, but especially the ones from Gym Leaders. This includes Chuck deciding to take his wife to Tohjo Falls, Falkner and Janine's war, Clair thinking her outfit is the latest fashion, and Whitney rambling on about nothing.
In particular, one of Chuck's phone calls has him talking about how his waterfall-generator has been malfunctioning, only to be cut off by it breaking and gushing water everywhere.
This piece of Lampshade Hanging... the man before the Elite Four states that he would offer to teleport you back to New Bark Town, but no one takes him up on his offer because everyone wanting to go there just flew there.note In the original Generation II games, you can't fly from Indigo Plateau to New Bark Town, making it necessary to have him teleport you there if you need to go back. Since flying was a more convenient option in the remakes, the option to have him teleport you back to New Bark Town is removed entirely. It doesn't help (or, for the player character, hurt) in the remakes that the Indigo Plateau and Victory Road are the only places in the entire game from where a person can fly to and from any city/town in either Johto or Kanto.
From the original Generation II games (but not the Generation IV remakes)...we have this infamous line:
From the remakes, when Lance suggests a tag team battle between you, Silver, himself, and his cousin Clair... for some reason, Silver really isn't too keen on the idea of Clair being on his team...or Ethan/Lyra being on a team with Lance. He immediately scoffs at Clair, insulting her wardrobe choice, then telling the hero character (regardless of gender), that you are on his team. We all know what kind of game Pokémon is, but many people couldn't help but find Silver acting like a jealous tsundere. It came off as rather hilarious.
Despite gender, the scene where Silver strips your character of their Team Rocket disguise...mostly because of the dozens and dozens of fanarts that have spawned from this one scene!
Just the simple fact that by wearing their uniform, Team Rocket members are willing to believe a preteen kid is part of their criminal group.
When you're battling Silver in the Radio Tower, he will say the following below when he's down to his last Pokemon. If you've seen The Dark Knight, then this comment will cause you to either snicker, giggle, snort, or a combination of all the above.
Silver: Hey...why so serious?
It's even funnier when his last Pokémon is his Haunter.
The foreign Team Rocket member in Cerulean City served little purpose other than to be a walking Funny Moment. Even though there are several Team Rocket grunts that match his level of ditziness, we dare you to find one with an accent that adorable.
Rocket Grunt: Oops! I so sorry! You not hurt, okay?
When you're disguised as a TR member during the Radio Tower Raid, and you talk to the Department Store Manager, he spurts out the following line note The other shoppers, as well as the Rocket grunts outside, will give you some pretty funny lines, too:
Department Store Manager: What kind of business would a Team Rocket member have here? We aren't selling sanity here.
Buena's line during the Radio Tower takeover:
Buena: Huh? Today's password? HELP, of course!
The guy at the Magnet Train Station
The train hasn't come in... I know! I'll carry the passengers on my back! That won't work.
The Gym guide in the Cerulean Gym, whose trainers include women in skimpy bikinis, says this out of all things.
Gym guide: Yo! Legend in the making! Since Misty was away, I went out for some fun, too. He-he-he.
In HG/SS when you talk to your Pokémon while getting ready to battle, they will look to one side then move half a step then look to the other side and move half a step, then the words on the screen will say:
Guide Words: Why is it doing warm-up stretches?
Your Pokemon react differently to different Kanto Gyms. Janine's has them hitting the invisible walls. Non-fire types appreciate getting a "shower" in Erika's gym.
In HG/SS, you can use your Pokédex to check the size and weight of a Pokémon compared to the player character. Weight is represented by a scale, holding the Pokémon and the player character in each of the cups. If the Pokémon is very heavy, the PC will bounce so high that they momentarily appear in the upper screen of the DS. If you use a Steelix or a Snorlax, the heaviest Pokémon available in a normal playthrough, the PC is catapulted straight through the upper screen and never comes back down.
In HeartGold and SoulSilver, on some occasions while your Pokémon is walking with you, it will hold items for the Fashion Case. There are a few points where if you try to take whatever it's holding, it will get angry at you because its item is apparently "too important to let go".
Stuff relating to the Day Care Center on Route 34:
Because the other player character, Lyra if you chose Ethan and vice versa, appears around Johto, we get this gem:
Ethan/Lyra: Grandma! Let me introduce my friend. This is [player name]!
Day Care Lady: Ah ha. So this is your... girl/boy...friend?
People used cheating devices to check if Joey is actually telling the truth, looking up the Rattata's stats to see if it has perfect Individual Values. Sadly, it's all zeros like other trainers of Joey's difficulty level.
There's a Pokefan on Kanto Route 13 who commands you to "bow down before his regal Pokemon". His team consists of a Nidoking and Slowking, both at much lower levels than anything else on the route and... a LEVEL 65 MAGIKARP!!! And it still has Splash. Getting an LPer's reaction to it live is bound to be hilarious, see one such example in this video.
This is actually an Incredibly Lame Pun that was Lost in Translation. In the Japanese versions, Nidoking has the same name, Slowking is Yadoking, and Magikarp is Koiking. They all have "king" in their names, hence regal.
Amusingly, the joke unintentionally turns into a Visual Pun instead because of Magikarp's crown-shaped dorsal fin.
This is only the case in the remakes though. The original English versions of the game replaced the Magikarp with a Seaking of a level similar to the other two Pokémon, preserving the pun, but ruining the fun of seeing an extremely high-levelled, yet still as useless, Magikarp.
As mentioned, the moment when your rival strips you out of your Team Rocket uniform. The Player Character, who usually acts like a Heroic Mime and doesn't respond to the things around them at all, even turns around to face the wall in a desperate effort to keep their rival from blowing their cover.
On route to the Safari Zone, non-Flying Types are scared of the cliff edge, and may get up close to you to avoid it. Non Fire-types will splash and play in the waves at any beach, while Fire Types will get grouchy and displeased.
On a related note, all Flying-Types will "fly majestically through the air" in certain areas... even stuff like Gyarados that doesn't look like it can fly!
On the outskirts of Blackthorn Town, there is a Black Belt NPC who, if you speak to him, says, "Aaaaoooo! Voices in my head!" He then clarifies that he's listening to his radio, but that doesn't really explain much.
Lance's sense of humor. At a point during Team Rocket headquarters section, you run into Lance standing next to a Rocket grunt:
Lance: It takes two passwords to get into the boss's quarters. Those passwords are known only to a few Rockets. That Rocket there very graciously told me so. Let's go get the passwords.