which areas of St Katherine's Monastery are closed?

It most certainly is.

The monastery is a real, working monastery and the monks graciously allow thousands of tourists to invade their privacy in order to see the splendour of the place. Most of the monastery is closed off but one does see the well, the main chapel, the supposed site of the burning bush, the chamber of skulls and an excellent museum which has some, historically speaking, incredible sights. Even when closed to the public, the exterior and the surrounding landscape are a marvel. I adore the place though its beauty is shattered when too many of the tour buses come up from Sharm el Sheikh.