[TotalPotato]

Bulbasaur: Fury Cutter, Snore
Ivysaur: Fury Cutter, Snore
Venusaur: Fury Cutter, Snore
Charmander: Fire Punch, Fury Cutter, Snore
Charmeleon: Fire Punch, Fury Cutter, Snore
Charizard: Fire Punch, Fury Cutter, Snore
Squirtle: Icy Wind, Dive, Ice Punch, Snore
Wartortle: Icy Wind, Dive, Ice Punch, Snore
Blastoise: Icy Wind, Dive, Ice Punch, Snore
Beedrill: Fury Cutter, Snore
Pidgey: Snore
Pidgeotto: Snore
Pidgeot: Snore
Rattata: Icy Wind, Snore
Raticate: Icy Wind, Snore
Spearow: Snore
Fearow: Snore
Ekans: Snore
Arbok: Snore
Pikachu: Snore
Raichu: Snore
Sandshrew: Snore
Sandslash: Snore
Nidorina: Snore
Nidoqueen: Icy Wind, Fire Punch, Fury Cutter, ThunderPunch, Snore
Nidorino: Snore
Nidoking: Icy Wind, Fire Punch, Fury Cutter, ThunderPunch, Snore
Clefairy: Icy Wind, Fire Punch, Ice Punch, ThunderPunch, Snore
Clefable: Icy Wind, Fire Punch, Ice Punch, ThunderPunch, Snore
Vulpix: Snore
Ninetales: Snore
Jigglypuff: Icy Wind, Fire Punch, Ice Punch, ThunderPunch, Snore
Wigglytuff: Icy Wind, Fire Punch, Ice Punch, ThunderPunch, Snore
Zubat: Snore
Golbat: Snore
Oddish: Snore
Gloom: Snore
Vileplume: Snore
Paras: Fury Cutter, Snore
Parasect: Fury Cutter, Snore
Venonat: Snore
Venomoth: Snore
Diglett: Snore
Dugtrio: Snore
Meowth: Icy Wind, Snore
Persian: Icy Wind, Snore
Psyduck: Dive, Ice Punch, Snore
Golduck: Dive, Fury Cutter, Ice Punch, Snore
Growlithe: Snore
Arcanine: Snore
Mankey: Fire Punch, Ice Punch, ThunderPunch, Snore
Primeape: Fire Punch, Ice Punch, ThunderPunch, Snore
Poliwag: Icy Wind, Dive, Snore
Poliwhirl: Icy Wind, Dive, Ice Punch, Snore
Poliwrath: Icy Wind, Dive, Ice Punch, Snore
Abra: Snore
Kadabra: Snore
Alakazam: Snore
Machop: Snore
Machoke: Snore
Machamp: Snore
Bellsprout: Snore
Weepinbell: Snore
Victreebel: Snore
Tentacool: Icy Wind, Dive, Snore
Tentacruel: Icy Wind, Dive, Snore
Geodude: Fire Punch, Snore
Graveler: Fire Punch, Snore
Golem: Fire Punch, Fury Cutter, Snore
Ponyta: Snore
Rapidash: Snore
Slowpoke: Icy Wind, Dive, -Ice Punch
Slowbro: Icy Wind, Dive, Fury Cutter, Ice Punch
Magnemite: Snore
Magneton: Snore
Farfetch'd: Snore
Doduo: Snore
Dodrio: Snore
Seel: Snore
Dewgong: Snore
Grimer: Fire Punch, Ice Punch, ThunderPunch, Snore
Muk: Fire Punch, Ice Punch, ThunderPunch, Snore
Shellder: Icy Wind, Dive, Snore
Cloyster: Icy Wind, Dive, Snore
Gastly: Snore
Haunter: Snore
Gengar: Snore
Onix: Snore
Drowzee: Snore
Hypno: Snore
Krabby: Icy Wind, Dive, Fury Cutter, Snore
Kingler: Icy Wind, Dive, Fury Cutter, Snore
Voltorb: Snore
Electrode: Snore
Exeggcute: Snore
Exeggutor: Snore
Cubone: Icy Wind, Fire Punch, ThunderPunch, Snore
Marowak: Icy Wind, Fire Punch, ThunderPunch, Snore
Hitmonlee: Snore
Hitmonchan: Snore
Lickitung: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Koffing: Snore
Weezing: Snore
Rhyhorn: Icy Wind, Snore
Rhydon: Icy Wind, Fire Punch, Fury Cutter, ThunderPunch, Snore
Chansey: Icy Wind, Snore
Tangela: Snore
Kangaskhan: Icy Wind, Fire Punch, Fury Cutter, Ice Punch, ThunderPunch, Snore
Horsea: Icy Wind, Dive, Snore
Seadra: Icy Wind, Dive, Snore
Goldeen: Icy Wind, Dive, Fury Cutter, Snore
Seaking: Icy Wind, Dive, Fury Cutter, Snore
Staryu: Icy Wind, Dive, Snore
Starmie: Icy Wind, Dive, Snore
Mr. Mime: Fire Punch, Ice Punch, ThunderPunch, Snore
Scyther: Snore
Jynx: Icy Wind, Snore
Electabuzz: Snore
Magmar: Snore
Pinsir: Fury Cutter, Snore
Tauros: Icy Wind, Snore
Gyarados: Icy Wind, Dive, Snore
Lapras: Icy Wind, Dive, Snore
Eevee: Snore
Vaporeon: Icy Wind, Dive, Snore
Jolteon: Snore
Flareon: Snore
Porygon: Icy Wind, Snore
Omanyte: Icy Wind, Dive, Snore
Omastar: Icy Wind, Dive, Snore
Kabuto: Snore
Kabutops: Dive, Fury Cutter, Snore
Aerodactyl: Snore
Snorlax: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Articuno: Icy Wind, Snore
Zapdos: Snore
Moltres: Snore
Dratini: Icy Wind, Snore
Dragonair: Icy Wind, Snore
Dragonite: Icy Wind, Dive, Fury Cutter, Ice Punch, Snore
Mewtwo: Icy Wind, Fire Punch, Ice Punch, ThunderPunch, Snore
Mew: Icy Wind, Dive, Fire Punch, Fury Cutter, Ice Punch, ThunderPunch, Snore
Chikorita: Fury Cutter, Snore
Bayleef: Fury Cutter, Snore
Meganium: Fury Cutter, Snore
Cyndaquil: Snore
Quilava: Fury Cutter, Snore
Typhlosion: Fire Punch, Fury Cutter, ThunderPunch, Snore
Totodile: Icy Wind, Dive, Snore
Croconaw: Icy Wind, Dive, Fury Cutter, Snore
Feraligatr: Icy Wind, Dive, Fury Cutter, Snore
Sentret: Fire Punch, Fury Cutter, Ice Punch, ThunderPunch, Snore
Furret: Fire Punch, Fury Cutter, Ice Punch, ThunderPunch, Snore
Hoothoot: Snore
Noctowl: Snore
Ledyba: Ice Punch, ThunderPunch, Snore
Ledian: Ice Punch, ThunderPunch, Snore
Spinarak: Snore
Ariados: Snore
Crobat: Snore
Chinchou: Dive, Snore
Lanturn: Dive, Snore
Pichu: Snore
Cleffa: Icy Wind, Snore
Igglybuff: Icy Wind, Snore
Togepi: Snore
Togetic: Snore
Natu: Snore
Xatu: Snore
Mareep: Snore
Flaaffy: Fire Punch, ThunderPunch, Snore
Ampharos: Snore
Bellossom: Snore
Marill: Icy Wind, Dive, Ice Punch, Snore
Azumarill: Icy Wind, Dive, Ice Punch, Snore
Sudowoodo: Fire Punch, Ice Punch, ThunderPunch
Politoed: Icy Wind, Dive
Aipom: Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Wooper: Dive, Ice Punch
Quagsire: Dive, Ice Punch
Murkrow: Icy Wind
Slowking: Icy Wind, Dive, Fury Cutter, Ice Punch
Snubbull: Fire Punch, Ice Punch, ThunderPunch
Granbull: Fire Punch, Ice Punch, ThunderPunch
Qwilfish: Icy Wind, Dive
Heracross: Fury Cutter
Sneasel: Fury Cutter
Teddiursa: Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Ursaring: Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Remoraid: Dive
Octillery: Dive
Delibird: Icy Wind
Mantine: Icy Wind, Dive
Kingdra: Icy Wind, Dive
Porygon2: Icy Wind
Smoochum: Icy Wind
Miltank: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Blissey: Icy Wind
Suicune: Icy Wind, Dive
Tyranitar: Fire Punch, Fury Cutter
Lugia: Icy Wind, Dive
Treecko: Fury Cutter, ThunderPunch
Grovyle: ThunderPunch
Sceptile: ThunderPunch
Combusken: Fire Punch, Fury Cutter, ThunderPunch
Blaziken: Fury Cutter, ThunderPunch
Mudkip: Icy Wind, Dive
Marshtomp: Icy Wind, Dive, Ice Punch
Swampert: Icy Wind, Dive, Ice Punch
Zigzagoon: Icy Wind, Fury Cutter
Linoone: Icy Wind, Fury Cutter
Lotad: Icy Wind
Lombre: Icy Wind, Dive, Fire Punch, Ice Punch, ThunderPunch
Ludicolo: Icy Wind, Dive, Fire Punch, Ice Punch, ThunderPunch
Nuzleaf: Fury Cutter
Shiftry: Fury Cutter
Wingull: Icy Wind
Pelipper: Icy Wind
Ralts: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Kirlia: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Gardevoir: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Surskit: Icy Wind
Masquerain: Icy Wind
Breloom: Fury Cutter, ThunderPunch
Slakoth: Icy Wind, Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Vigoroth: Icy Wind, Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Slaking: Icy Wind, Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Nincada: Fury Cutter
Shedinja: Fury Cutter
Whismur: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Loudred: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Exploud: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Makuhita: Fire Punch, Ice Punch, ThunderPunch
Hariyama: Fire Punch, Ice Punch, ThunderPunch
Azurill: Icy Wind
Nosepass: Fire Punch, Ice Punch, ThunderPunch
Skitty: Icy Wind
Delcatty: Icy Wind
Sableye: Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Mawile: Icy Wind, Ice Punch, ThunderPunch
Aron: Fury Cutter
Lairon: Fury Cutter
Aggron: Icy Wind, Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Plusle: ThunderPunch
Minun: ThunderPunch
Volbeat: Ice Punch, ThunderPunch
Illumise: Ice Punch, ThunderPunch
Roselia: Fury Cutter
Gulpin: Fire Punch, Ice Punch, ThunderPunch
Swalot: Fire Punch, Ice Punch, ThunderPunch
Carvanha: Icy Wind, Dive, Fury Cutter
Sharpedo: Icy Wind, Dive, Fury Cutter
Wailmer: Icy Wind
Wailord: Icy Wind
Spoink: Icy Wind
Grumpig: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Spinda: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Flygon: Fire Punch
Cacnea: Fury Cutter, ThunderPunch
Cacturne: Fury Cutter, ThunderPunch
Zangoose: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Seviper: Fury Cutter
Barboach: Icy Wind, Dive
Whiscash: Icy Wind, Dive
Corphish: Icy Wind, Fury Cutter
Crawdaunt: Icy Wind, Dive, Fury Cutter
Feebas: Icy Wind, Dive
Milotic: Icy Wind, Dive
Castform: Icy Wind
Kecleon: Icy Wind, Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Shuppet: Icy Wind
Banette: Icy Wind
Duskull: Icy Wind
Dusclops: Icy Wind
Tropius: Fury Cutter
Chimecho: Icy Wind
Absol: Icy Wind, Fury Cutter
Spheal: Icy Wind, Dive
Sealeo: Icy Wind, Dive
Walrein: Icy Wind, Dive
Clamperl: Icy Wind, Dive
Huntail: Icy Wind
Gorebyss: Icy Wind
Relicanth: Icy Wind
Luvdisc: Icy Wind, Dive
Bagon: Fury Cutter
Shelgon: Fury Cutter
Salamence: Fury Cutter
Metang: Icy Wind, Fury Cutter, Ice Punch, ThunderPunch
Metagross: Icy Wind, Fury Cutter, Ice Punch, ThunderPunch
Regirock: Fire Punch, Ice Punch, ThunderPunch
Regice: Ice Punch, ThunderPunch
Registeel: Ice Punch, ThunderPunch
Latias: Icy Wind, Dive, Fury Cutter
Latios: Icy Wind, Dive, Fury Cutter
Kyogre: Icy Wind, Dive
Groudon: Fire Punch, Fury Cutter, ThunderPunch
Rayquaza: Icy Wind, Dive, Fury Cutter
Jirachi: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Deoxys: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Roserade: Fury Cutter
Ambipom: Fire Punch, Fury Cutter, Ice Punch, ThunderPunch
Munchlax: Fire Punch
Weavile: Fury Cutter
Lickilicky: Icy Wind, Fire Punch, Ice Punch, ThunderPunch
Rhyperior: Icy Wind, Fire Punch, ThunderPunch
Mamoswine: Fury Cutter
Porygon-Z: Icy Wind
Gallade: Fire Punch, Ice Punch, ThunderPunch
Probopass: Fire Punch, Ice Punch, ThunderPunch
Dusknoir: Icy Wind

All of the above moves are 4th gen tutor moves that were either tutor moves or HM moves in the 3rd gen. I checked against Veekun and Serebii, and when the lists conflicted (Mamoswine learning Fury Cutter, Slowpoke not learning Ice Punch), I checked it in my own HG game (in both cases, Serebii turned out to be correct).

The Smogon list notably excludes the Pokémon that learnt the move both in the 3rd gen and in the 4th gen, but include the Pokémon that only learns the move in the 4th gen (except alternate forms of Deoxys). 4th gen evolutions of non-4th gen Pokémon also obey this rule, but pre-evolutions do not (however, Munchlax did not have Fire Punch, although it did have Ice Punch). I don't know why these trends apply, and I don't know whether there's an easy fix that doesn't involve me trawling through the other tutor moves, but I thought it deserved a mention.

If there isn't an easy fix, I'll probably update this list with more tutor moves later (or should I make a new post in case it gets confusing with C&C mods checking lists and such?).