The following can appear as shiny but can't be caught, as well as the games they appear in:
Fire Red/Leaf Green:
- the weedle encountered during the catching tutorial
- another weedle in the Teachy TV catching tutorial
- the Marowak that appears in the Pokemon tower in Lavender town (note it won't appear shiny appear until the silph scope is used)
Ruby/Sapphire/Emerald:
- the poocheyena that attacks Professor Birch at the start of the game in Ruby/Sapphire
- the zigzagoon that attacks professor Birch at the start of the game in Emerald
- both the wild raltz and Norman's zigzagoon during the catching tutorial (note Raltz will not be shiny in subsequent battles)
- Both the wild Pokemon in the battle pike and battle pyramid (Emerald only)
- the Pokemon you rent from the battle factory (Emerald only)
Diamond/Pearl/Platnium:
- the starly that attacks at the beginning on the game
HeartGold/SoulSilver:
- the Pokemon used by the AI in the Battle Frontier, this can also occur in Platnium
Black/White & Black 2/White2:
- the Minccino Professor Juniper toose out in the opening sequence (Black/White)
- the Cinccino Professor Juniper tosses out in the opening sequence (Black 2/White 2)
- the wild purrloin and bianca's lillipup during the catching tutorial (Black 2/White 2)
X/Y
- the wild bunnelby and Calem/Serena's fletchling can both be shiny
Omega Ruby/Alpha Sapphire:
- the raltz wally catches may be shiny but the only evidence I could find was this reddit thread, so take it with a grain of salt
Sun/Moon
- an NPC will battle you with a shiny exeggcute in Sea Folk Village in the Pokemon Centre
Sword/Shield
- in the Isle of Armour DLC an NPC can show you a Pokedex entry which has a chance of being shiny
- a GMAX Urshifu was uncatchable in a raid battle event held in June 2022 as seen in this video
- high level Pokemon unable to be caught when the player doesn't have enough gym badges will be shiny locked as seen in this reddit thread
Brilliant Diamond/Shining Pearl:
- like the originals, the wild starly that attacks at the beginning of the game can also be shiny
- the bidoof Dawn/Lucas captures in the catching tutorial
Pokemon XD Gale of Darkness:
- every Pokemon the player uses in battle CDs as you only have them for the length of the battle
Lastly, any wild Pokemon encountered before the player obtained Pokeball will be uncatchable, such as in Gold/Silver/Crystal
I could not find any evidence there are unobtainable shiny Pokemon in Ultra Sun/Ultra Moon, Let's Go Pikachu, Let's Go Eevee or Legends: Arceus
Main Source
Hope this helps
