In the Japanese Pokémon Red and Green, if you evolve your starting Pokémon before you get the Pokédex (raising it from level 5 to 16 against Pokémon about levels 2 through 4 before you get to the second town and back), then you can never get the Pokédex. This means you can't get Pokéballs, and that means you can't get past the second town, Viridian City. This was corrected in Pokémon Blue, as well as all international releases.
Note that if you try to trade your starter to another game to give it Cut to get around that man, the badge necessary is further up in your adventure.
Gen I has the famous Glitch City, which - if the player follows a particular sequence of actions involving the Safari Zone - will put you in a "town" made of a random jumble of tiles pulled from the town you last visited. Walk too far North, South, or East, and the game crashes. Walk too far West and... you lose the ability to go back, and then the game crashes. The only way out is to Fly to another city or Teleport/Dig yourself back to the last Pokémon Center you visited. If you save in Glitch City without a Pokémon who knows one of those moves, you are in trouble.
Also in Gen I, the glitch Pokémon other than Missingno., such as 'M. You usually have to go out of your way to encounter them, especially in Yellow. But the harder you have to look, the more damage they do. Missingno. rarely does anything worse than mess with your Hall of Fame; other glitched Pokémon and trainers can screw up your party, destroy your savegame, or even render the entire cartridge unusable. Demons in all their glitchy glory here.
Going to catch Articuno? Make sure to bring an Escape Rope or a Pokémon that knows Dig with you! Otherwise, once you push those boulders into place, you're trapped forever.
It's possible to get the Machine Part before you fight the foreign Team Rocket member on the Nugget Bridge. Doing this causes the man at the Power Plant to not notice that you have it, which prevents you from using the Magnet Train or getting the EXPN Card which is required to wake the Snorlax in Vermillion City.
Early Japanese copies of Diamond and Pearl have two for the price of one at the Pokémon League. Both were fixed in later Japanese releases and all international versions.
The first one is the infamous "Surf through Aaron's door" glitch used to access Newmoon Island and Flower Paradise. If you saved in the black abyss while the game thinks you're somewhere where you can't use Fly or Teleport (or just don't have either), you're stuck. Keep in mind that getting Darkrai and Shaymin with this glitch requires saving the game.
The second is one you're more likely to come across completely by accident. All Pokémon Centers have a second floor where local wireless activities are conducted, the Pokémon League being no exception. If you go to this second floor at the Pokémon League, the escalator to bring you back to the ground floor doesn't work, leaving you trapped there unless you load your previous save. Given what the room is for, your last save was probably there. Players who managed to get stuck here could send their games in to Nintendo to get hacked out of the room, or use Action Replay codes to do so themselves.
An oversight in the early Japanese copies of Diamond and Pearl makes it possible for the player to go to the Underground on a bridge tile in Sunnyshore City. When entering the Underground, the player is required to save the game. The problem with this is, when going into the Underground on that particular bridge tile, then going up into the normal world again, the player character will land UNDERNEATH the bridge tile and be able to walk on out-of-bounds sea tiles (and, among other funny things, can actually stand IN the Munchlax-shaped rock). Because of having saved, you will be stuck if you didn't bring a Pokémon knowing Fly or Teleport with you.
In Pokémon Black and White, a minor but annoying instance of this can occur. It's unclear exactly what causes it, but occasionally a Pokémon will survive an attack with 1 HP but still faint. All attempts to proceed with the battle will fail, and if this occurs in a trainer battle, the only way to get out is to restart the game. Luckily, if this happens in the Battle Subway or against a live person, you can still end the battle through forfeit.
In that same game there was also a Battle Video of someone fighting an NPC; the NPC fisherman's Walrein fainted, and after getting his next Pokemon knocked out, he sent out Walrein again; its HP bar didn't appear, and the game acted as if the trainer's Pokemon was attacking nothing.
In the Japanese version of Pokémon Black 2 and White 2 it is possible to exit Victory Road during the Zoroark event, making it impossible to access a certain Bonus Boss. This was fixed in the international releases.