I am not going to include any circumstances where Pokemon know moves that they shouldn't in spin off games.
Lance's Dragonite knows Barrier. (Red, Green, Blue) (Dragonite cannot learn Barrier until Gen VI.)
Lance's Aerodactyl knows Rock Slide. (Gold, Silver, Crystal) (Aerodactyl cannot learn Rock Slide until Gen III.)
Cooltrainer Dianne's Lanturn knows Earthquake. (Emerald) (Lanturn cannot learn Earthquake.)
Totodile in the Battle Hall know Brine. (Platinum, HeartGold, SoulSilver) (Totodile cannot learn Brine.)
Roselia in the Battle Hall know Sludge. (Platinum, HeartGold, SoulSilver) (Roselia cannot learn Sludge.)
Anorith in the Battle Hall know Stone Edge. (Platinum, HeartGold, SoulSilver) (Anorith cannot learn Stone Edge.)
School Kid Ann's Wormadam knows Leaf Storm. (Black and White) (Only Plant Cloak Wormadam can learn Leaf Storm, while School Kid Ann's Wormadam is Sandy Cloak.)
En's Moltres knows Leer. (Black 2 and White 2) (Moltres can only learn Leer in Gen I, but it cannot be transferred to Gen V.)
Abel's Metagross knows Ice Beam. (Black 2 and White 2) (This was an error, as it was instead meant to have Ice Punch.)
Alomomola called by Totem Wishiwashi can know Water Gun. (Sun and Moon) (Alomomola cannot learn Water Gun.)
Spearow in the Battle Hall know Quick Attack and Scary Face. (Platinum, HeartGold, SoulSilver) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Drowzee in the Battle Frontier know Assist and Barrier. (Platinum, HeartGold, SoulSilver) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Gligar in the Battle Frontier know Sand Tomb and Metal Claw. (Platinum, HeartGold, SoulSilver) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Kangaskhan in the Battle Subway know Double-Edge and Circle Throw. (Black, White, Black 2, White 2) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Gothitelle in the Battle Subway know Mirror Coat and Miracle Eye. (Black, White, Black 2, White 2) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Walter's Parasect knows Bullet Seed and Leech Seed. (Black 2 and White 2) (Unable to have both of these moves at the same time because of Bullet Seed being a TM in Gen III and IV but not Gen V.)
Janine's Crobat knows Hypnosis and Brave Bird. (Black 2 and White 2) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Bryon's Aggron knows Head Smash and Dragon Rush. (Black 2 and White 2) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Winona's Tropius knows Leaf Blade and Curse. (Black 2 and White 2) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Burgh's Accelgor knows Guard Split and Spikes. (Black 2 and White 2) (Unable to have both of these moves at the same time because of the way egg moves worked.)
Pryce's Piloswine knows Fury Attack. (Gold, Silver, Crystal) (Pryce's Pilowswine is on level 31, yet Piloswine cannot learn Fury Attack until level 33.)
Sidney's Shiftry knows Extrasensory. (Ruby, Sapphire, Emerald) (Sidney's Shiftry is on level 48, yet it would have to reach level 49 as a Nuzleaf and then be evolved to Shiftry in order to have Extrasensory.)
Winona's Altaria knows Dragon Dance. (Ruby, Sapphire, Emerald) (Winona's Altaria is on level 33, yet Altaria cannot learn Dragon Dance until level 40 in Gen III.)
Norman's Linoone knows Slash. (Emerald) (Norman's Linoone is on level 29, yet Linoone cannot learn Slash until level 41.)
Bianca's Purrloin knows Growl. (Black 2 and White 2) (Bianca catches a wild Purrloin on level 2 during the capture tutorial, yet Purrloin cannot learn Growl until level 3. Purrloin can have Growl on level 2 if breeding is used, but wild level 2 Purrloin cannot have Growl.)
Wulfric's Cryogonal knows Confuse Ray. (X and Y) (Wulfric's Cryogonal is at level 40, yet Cryogonal cannot learn Confuse Ray until level 45.)
Veteran Brenden's Hippodon knows Dig. (Platinum) (Hippodon cannot learn Dig in Platinum, however this was changed when HeartGold and SoulSilver were released.)
Viola's Masquerain knows Air Slash. (X and Y) (Viola's Masquerain is on level 40, yet Masquerain cannot learn Air Slash until level 47 in X and Y.)
Source
EDIT: I've added the reasons as to why each Pokemon listed above has illegal move(s).
