Personally, I think the biggest issue is that in supernatural belief systems the truth is relative. The truth can be whatever you believe it to be. In this way, I think some belief systems rob the truth of its inherent meaning. This idea, for me anyway, defines the inherent problem with believing in the supernatural.
For example, if you want to believe that someone named Noah built an ark and put every animal on the planet in pairs on the boat to save them from a catastrophic flood then its perfectly acceptable and even encouraged. This is all despite the fact that this supposedly happened in the middle east somewhere although there is not a single shred of credible evidence to support it. Strange, considering how the middle east is among the most heavily archaeologically excavated areas in the world--so you'd think we would have found some piece of evidence in favor of it.
This example is fairly benign, but you can see how this type of logic would cause problems. How do people separate what they want to believe from what is actually true. What about when parents believe that vaccines are dangerous and cause autism, despite the evidence to the contrary, and don't get their kids vaccinated and start a measles outbreak in their community? What about when policymakers block funding for stem cell research simply because they believe that zygotes have souls...despite the complete lack of evidence that souls even exist.