The controller of Protector of the Sanctuary is not prevented from activating card effects that allow him to draw, but he may not activate a card effect that includes both players, like Card Destruction.
The opponent may still activate cards that do not immediately cause him to draw, like Mirage of Nightmare, because the effect has not activated/triggered yet and the card is a continuous effect. He will be unable to draw when the effect does trigger, as long as Sanctuary is face-up.
You cannot activate Pot of Greed since it causes you to draw 2 cards immediately.
The controller of Sanctuary cannot activate Disturbance Strategy or other cards that force his opponent to draw cards.
The only time your opponent can respond to the effect of a card that causes him to draw is a monster effect, but he still cannot draw cards if that is a requirement, like with Sasuke Samurai #3, Bistro Butcher, Morphing Jar.