Minato Hokage is an old ninja, therefore his programming does not count Irremovable Tag as ''Tag'', because when he was created, there isn't an ''Irremovable Tag'' to begin with.

So Irremovable Tag will not activate field mystery of Minato.

As for new ninjas in the future, they will count both debuff types in their programming. Hidan will count Irremovable Tag' and Tag as ''Tag'', Indra and Deidara Inception will count Irremovable Ignition and normal Ignition as ''Ignition''.

Also additional knowledge, Ay 4th is an old ninja as well, so he doesn't count Irremovable Tag as ''Tag'', so he will not trigger his passive 1 when there's Irremovable Tag present.