A lot of places have already been closed to tourists because of what the tourists have done to these places. For example, some tourists vandalize the place because they think that it is cool to have their names placed there. Some tourists just litter all over the place and it would take some years of rehabilitation to improve everything.
At times, natural causes may be the reason for the place closing down. Snake Island is already closed because of the species of snakes that can be found there. There are also other places that closed down because there were not enough people going there.
There are many places in the world that have been closed down to locals and tourists alike. Sometimes it is because the attraction has fallen into disarray, others close because of an accident, and still others just don't have the number of guests needed to make it sustainable. Interestingly enough, Snake Island off of the coast of Brazil is closed to the public due to a species of snake that lives on the island.
This snake's venom is flesh-eating and the serpent has just about overrun this piece of land making it very dangerous for visitors. The North Sentinel Island is also closed to the public due to the natives that live there. They do not want contact with the outside world and have killed people that have been unlucky enough to be stranded there.