Basically, if the IF doll doesn't become real through the potion, it's attached permanently to your sim. It is out of the player's control -- you can't sell it, and as I found, you can't leave it behind when you move. I think someone said you can have your sim bury it in the sandbox as a child, but I haven't tried that.
In my sim's case, the IF doll is inert, that is, it never "grew up" but just stayed a doll. I guess you could say it's okay to take the doll with you whether inert or active, provided you don't make it real through the potion after the sim moves.