Impossible, or at least highly unlikely. I cannot imagine a plausible mechanism by spraying water up the nose, which is basically what ZICAM's activity is. It is a homeopathic remedy, meaning that there is no active ingredient. To claim harm would be as bankrupt as claiming effect!
The HI-larious part of the story is that Matrixx, the company that manufactures the water in a squirt bottle says, "there is no known causal link between the use of Zicam Cold Remedy nasal gel and impairment of smell. No well-controlled scientific study has demonstrated a potential cause-and-effect relationship between the use of Zicam and diminished smell function. No court cases have revealed any reliable evidence of any causal relationship."
You can say exactly the same about its ability to cure disease! Just substitute "treat a cold" for "impairment of smell" and you've got it right!