I think an Illusion who has been removed from the board effectively "doesn't exist", like a Godly Being who has not yet entered the board. Her abilities don't work and she can't trigger incidents even if she has enough paranoia and/or is Obstinate.
I'd like official confirmation though, since I could also see it being ruled that she still exists, just not in any of the four locations.
The only relevant rule I can find is this line from the Mastermind script creation guide:
Please note that even if she’s removed from the board due to her ability, she
returns to the board in the next loop.
Obviously this is redundant with the goodwill ability's text, but it seems extra redundant if while removed from the board she can still participate in the loop. To me it seems to imply that while off the board she is removed from the game.