I have seen professors clearly stating that they do not accept visitor students or postdocs.
So, what are the disadvantages of having such positions?
This can vary a lot from professor to professor or even school to school. Some ideas:
Edit: If you're seeing this on a website, the professor may not have a standing rule against visitors or postdocs, but may simply not have room or funding for them right now.
Many professors receive a lot of unsolicited emails from people asking to come to their lab/department for a summer internship, sabbatical, collaboration visit of several days/weeks/months/years, etc. The professor to whose web page you linked seems to have gotten tired of these requests and therefore declared that he will not be taking on any visitors. I doubt his strategy has much effect, but that's why he's doing it. It has nothing to do with a disadvantage of having visitors, and everything to do with the disadvantage of receiving many tedious requests from people you don't know or have any interest in working with.