A shy man may never become courageous enough to approach women in general, but he can eventually grow comfortable with a girlfriend or wife.As the fear of the unknown dissipates with increased familiarity, a shy man's confidence in his interactions with you will rise.Once this happens, you will need to gradually assume a less assertive (more traditionally female) role, because he will then be more liable to get bored with you, wonder if he could do better, consider his other options (the perception of which his newly-found confidence will likely inflate), fool around, etc.The second caveat is that even a shy man has pride.That offense will reflect negatively on you and the relationship.
So show him that you want something more - ask repeatedly.
The reason for this is that shy men are more concerned with avoiding the social catastrophe of asking you out and failing than they are of getting you.
This is what prevents them from taking the initiative in the first place.
Comments like "Oh, I thought you were going drive/get the tickets/plan the trip/[insert other traditionally masculine roles here]" can help in priming his confidence if it seems to be lagging and it bothers you.
Dating a shy man takes a lot of initial investment for an uncertain outcome.