Generally speaking, Frozen items do not have their standard passive effects active.
Note that a Frozen Marieke Doll will still give Wrath Of The Admin if removed from the game.
Note that a Frozen item which is "inherently" Coldblooded from its own effect will still gain double the turns of Frozen if it gains additional turns of Frozen.
A Scarf Of The Serpent with its passive effect active will be Frozen for double the amount of turns. However, an already Frozen Scarf Of The Serpent will NOT gain double the turns of Frozen if it gains additional turns of Frozen; this appears to be different from an item like a Crocodile since the Scarf's effect also affects other items.
An item can have a max. of 99 turns of Frozen stored up at once.