No, it's not completely forbidden. Though you shouldn't move through the crease on your own. It really depends on the referee. Just yesterday I saw a ref point out to a player while the play was still going on that he was in the crease - that was while he was standing right at the crease with a defenseman next to him - he moved his feet a bit and play went on without stoppage. Other refs stop the play right away or wait till a shot happens, only to stop play then.
IIHF rules allow for a goal to be scored if the forward was pushed into the crease by a defenseman or if a forward is standing in the crease and not interfering with the goalie in any way (that includes interfering with his vision).
A goal shall be allowed:
4. If a player of the attacking team has been physically interfered with, by the action of any defending player so as to cause him to be in the goal crease when the puck enters the goal net, unless if in the opinion of the Referee, he had sufficient time to get out of the crease,
5. If the puck should become loose in the goal crease and then put into the goal net by the stick of the attacking player,
7. If an attacking player being in the goal crease at the moment the puck crosses the goal line and in no way affects the goalkeeper's ability to make a save, unless the cases described in Rule 471.
471 - DISALLOWING A GOAL
a) No goal shall be allowed:
4. If an attacking player stands or holds his stick in the goal crease when the puck enters the goal net, unless he has been physically interfered with, by the action of any defending player so as to cause him to be in the goal crease when the puck enters the goal net, unless if in the opinion of the Referee, he had sufficient time to get out of the crease or unless Rule 470 applies.
6. If an attacking player initiates contact with the goalkeeper, incidental or otherwise, while the goalkeeper is in his goal crease and a goal is scored.
7. If an attacking player initiates any contact with the goalkeeper, other than incidental contact, while goalkeeper is outside his goal crease and a goal is scored.
8. Where an attacking player enters or takes a position within the goal crease, so as to obstruct the goalkeeper's vision and impair his ability to defend his goal net and a goal is scored.
Under IIHF rules, you cannot stand in the crease on your own accord when the puck is not there. You can skate through it (without interfering with the goaltender obviously), but if you skate into the crease and stay there (even without making contact with the goaltender), play is stopped and the faceoff goes to the neutral zone.