The site owner/admin can do whatever the Hell he/she wants to. The First Amendment only restricts Congress from making laws that interfere with a free press, just the Congress, nobody else. What has actually happened to it is that courts have extended the same rule to the Executive Branch and the States, as well as "Found" exceptions for things like libel and treason/sedition laws, so in application there are a lot of bells, whistles, and booby-traps that don't exist in the Amendment's plain language.
Of course the site owner does open up the door to being prosecuted for fraud, libel, or things of that sort if the post is edited to say something truly outrageous and there is no indication or notice that it was edited, and it is presented as something the poster really said. Given the anonymity of most discussion board monikers, it would be very difficult to prove any such mockery would actually harm a real and identifiable person, and damnede few prosecutors would want a piece of that action. There could be a civil liability issue for the same reasons, and with the same problems.