Can I prevent my opponent from looking at cards with Vizier of the Menagerie while I'm Mindslaving them?
Yes, the Mindslaver's controller may prevent the controlled player from taking that specific action.



Mindslaver's rulings say:


  You get to make every decision the other player would have made during that turn. You can’t make any illegal decisions or illegal choices — you can’t do anything that player couldn’t do. You can spend mana in the player’s mana pool only on that player’s spells and abilities. The mana in your mana pool can be spent only on your spells and abilities.


Since looking at the top card of your library is an action you may take while controlling Vizier of the Menagerie, the player controlling the turn may simply choose not to take this action.