In the Japanese Pokémon Red and Green, if you evolve your starting Pokémon before you get to the second town and back (raising it from level 5 to 16 by Level Grinding against Pokémon about levels 2 through 4), then you can never get the Pokédex. This means you can't get Poké Balls, 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 of Pokémon Red and Blue.
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. As Glitch Cities are not part of the game's map, they have no real exits (in some you can't even move off the tile you start in), thus 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 one without a Pokémon who knows one of those moves, you are in trouble.
Also in Gen I, the glitch Pokémon other than Missingno. and '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. and M rarely do 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.
You have to buy a drink to get past the guards. If you defeat all the trainers up to that point and then spend or blackout to lose all your money, there is no way to enter Saffron City. The only possible loophole is to trade with another game that has a Pokemon with the move Payday. This is corrected in FireRed and LeafGreen by having the guards respond to the key item Tea, instead of a regular item you have to buy. However, the change in the remakes is more due to the fact that items can be traded to other players in those games, as Trainers can be challenged to a rematch in them.
A more minor case, most infamously seen in Twitch Plays Pokémon Red, occurs inside Cinnabar Island's Pokémon Mansion. One rotating NPC trainer can spot the player while they are in a nearby one-tile-wide nook, walking up to them and challenging them to a battle. When the battle ends, the NPC will remain in place in front of the narrow space, blocking the player's only exit. Provided the player doesn't have an Escape Rope, Dig, or Teleport on them, the only ways to escape the Mansion when this occurs are to reset or battle wild Pokémon until losing and being returned to a Pokémon Center.
There comes a point where you need to travel on water, requiring HM03, Surf, which is found in the Safari Zone. It costs 500 Pokémon Dollars to go inside the Safari Zone. Don't have 500? You're stuck forever!
More of a metagame example, but if both you and your opponent send out Wobbuffet with Leftovers, the match is made unwinnable since Leftovers recover more than Struggle will hit (and recoil) for, and Shadow Tag will block switching out. The move and Ability have been modified in Generation IV to make this situation impossible.
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. Due to a weird glitch with EVs, 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 Pokémon knocked out, he sent out Walrein again; its HP bar didn't appear, and the game acted as if the trainer's Pokémon was attacking nothing.
Back when the games were only in Japanese, there was a similar issue with Vanilluxe.
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.
In version 1.0, if you save outside on the main streets of Lumiose City, there is a chance that, when you load the game back up, Lumiose City will outright fail to load: your character will be frozen, the music will not be playing, and the city textures will not load. This renders the game unplayable, and thus would have to start the whole game over. It does not help that Lumiose City is easily the most confusing and disorienting city in the series, so many players may save there to take a break and/or get their bearings. Luckily, players have found ways to be able to resume the game if affected, and thankfully Nintendo has since released a patch to fix this glitch.