why do you think it is forbidden for anyone but a priest to visit the dead places?

The phrasing of this question indicates that the answer is left up to reader opinion. A reader's opinion and reason for why it is forbidden to go east will likely change as he/she progresses through the story.
At the beginning of the story, a reader is likely to assume that going east is forbidden because of nothing more than superstition and tribal tradition. The opening paragraph of the story gives readers the impression that the story is about a people that believe in spirits and demons; therefore, they need medicine-man-type people called "priests." We are told about metal needing to be purified for some unknown reason, and John tells us that those are the rules and they are never questioned.  

These are the rules and the laws; they are well made. It is forbidden to cross the great river and look upon the place that was the Place of the Gods—this is most strictly forbidden. We do not even say its name though we know its name. It is there that spirits live, and demons . . .

As readers keep reading, we start to learn that there is something odd about the metal superstition. We are told that only priests and soon-to-be priests can hold metal without dying. Why would that be case? Readers are given an early hint when John mentions the "Great Burning." By the end of the story, it's clear to readers that the Great Burning was some kind of nuclear holocaust. That clears up a lot about the metal and its power to kill people. The metal would have been irradiated for many years, and anybody that came near it would suffer radiation sickness and possibly die. The Dead Places are former homes and buildings, and they are likely loaded with metals of various kinds. Keeping people away from those places isn't just superstition. The rule has probably saved the lives of many people. By not going to the Dead Places, they are not exposing themselves to irradiated metals.
It's likely that the nuclear event happened east of John's village in the former city of New York. There would likely have been a cloud of nuclear radiation hanging over/around New York for many years as well. Anybody that traveled east was likely to get sick and die. That's why Chernobyl is a giant exclusion zone. In the story, the ban on going east and into any of the Dead Places is more than simple superstition. It's a rule that was originally designed to keep people safe and save lives.