*Ignore it, I thought you meant something else*
There was a thing in Gymbaliar, where Jessie ordered Croagunk to use Brick Break, but the hand was purple *Or vice-versa, with Poison Jab and the white hand*
During Caterpie's battle with Pidgeotto in Ash Catches a Pokémon!, Ash tells Caterpie to use Counter; however, Caterpie cannot learn Counter, yet this doesn't stop it from attempting (and failing) to use it anyway.
In Ditto's Mysterious Mansion, Duplica tells her Ditto to use Constrict on Ash's Bulbasaur after Transforming into said Bulbasaur. Bulbasaur cannot learn Constrict by any means. However, Ditto obeyed, implying that Ash's Bulbasaur may know Constrict.
In Who Gets to Keep Togepi?, Misty gets stuck with Psyduck in her battle with Ash. Ash doesn't battle seriously and has Bulbasaur simply lick Psyduck's head and tickle it with its vines. This is not to be confused with the actual moves Lick and Tickle, the latter of which was later introduced in Generation III. Bulbasaur still cannot learn Lick or Tickle. It is worth noting, however, that Ash's specific words were "Use Tickle!".
In Bound for Trouble, Pikachu and Meowth fight against the giant Rhydon. When Meowth jumps on Rhydon's back, he states that he should try a Tickle attack, and then acts like he would. However, Meowth cannot learn Tickle in any generation yet, and as mentioned with Bulbasaur, Tickle was not introduced until Generation III.
In Playing with Fire!, a Steelix uses Wrap. Steelix cannot learn Wrap, but it can learn Bind (a slightly less accurate version of Wrap with the same effect).
In Winner By a Nosepass!, Roxanne commands her Nosepass to use Hyper Beam which it cannot in the game. However in the dub these errors were picked up on, and the references to Hyper Beam were replaced with Thunder Wave and Zap Cannon (which can be learned by Nosepass).
In Second Time's the Charm, Anabel's Espeon uses Zap Cannon, which, though impossible to have in Generation III or IV, was available to teach to Espeon by TM07 in Generation II.
In Drifloon on the Wind!, some Drifloon were used to fly around the area. However, they did not use the actual move Fly (which is just as well, as they are unable to learn said move). The Pokédex specifically states that Drifloon can't carry the weight of children anyway. It is, however, able to carry players across short distances in Pokémon Ranger: Shadows of Almia.
A group of Combee in An Angry Combee-nation! appear to be using Psybeam; however, this is actually the anime representation of Vespiquen's Attack Order.
In Journey to the Unown!, the fact that Bronzor cannot learn Teleport was also picked up by the producers of the dub, and the appearance of it using Teleport was cut from the episode.
Pokémon Ranger and the Kidnapped Riolu! (Part 1) and Pokémon Ranger and the Kidnapped Riolu! (Part 2) feature a Riolu that can use Aura Sphere. This is due to its unique ability to manipulate aura prior to evolution. This episode is directly based on a mission from Pokémon Ranger: Shadows of Almia featuring a similar Riolu. This is the only way a Riolu that knows Aura Sphere is available in the games.
