Of course there is an easy way for the clerical regime in Tehran to put a stop to the current hysteria. But the ayatollah has yet to appear to declare that the reports of his death are exaggerated. Until he does, the chances are the rumours will spread.
In addition, the article takes a few shots at Michael Ledeen, the primary source for the rumors, saying he has a "track record in spreading misinformation" and has falsely reported Khamenei's death before. The Guardian also cites a Vanity Fair article that linked Ledeen to the yellowcake uranium intelligence fiasco.
The Guardian somewhat misleads on the latter point. While Vanity Fair provides plenty of circumstantial evidence, it also admits "there are no fingerprints connecting Ledeen to the Niger documents. Even his fiercest adversaries will concede this." And Ledeen does have many adversaries (for example, see here or here).
Furthermore, Ledeen himself admits he has misreported Khamenei's death before. He did so in the original post and reiterates that fact in a new post. In Ledeen's words, "I have never claimed to know anything about Khamenei's death, and I still don't know anything."
Of course, such details get lost in a 140 character Twitter message. Before you know it, conjecture is truth and rumor is news.
Maybe Ledeen should've known that before blogging. But then again, everyone spreading the rumor should've known that too before drinking the Kool-Aid.