There's two ways a Pokémon can be impossible to obtain as a Shiny. Either it's restricted to an Event (which usually never appear Shiny) or the Pokémon is only obtainable in-game, but is Shiny Locked there.
With that criteria, the following Pokémon can not be obtained in their Shiny forms through any legitimate means:
Ash Hat Pikachu (this includes every single form ranging from Kanto, Johto, Hoenn etc.)
Victini
Keldeo
Meloetta
Ash-Greninja (regular Greninja can still be Shiny, but the Event Greninja with Battle Bond is Shiny locked)
Hoopa
Volcanion
Cosmog
Cosmoem
Magearna
Eternatus
Kubfu
Urshifu
Zarude
Glastrier
Spectrier
Calyrex
Enamorus
Vivillon (Fancy Pattern)
Vivillon (Poké Ball Pattern)
Moltres-Galar
Dialga (Origin Forme)
Palkia (Origin Forme)
Source
Hope this helps!