From what I can tell, there might not even be any significant difference between the effect of a normal and an "ionic" hair drier. From looking at the claims of manufacturers, the claims and explanations of "ionic" hair driers are sketchy.

From what I can find, there are no health effects from using an ionic hair dryer (minus the normal hazards). It seems to emit the same amount of ions we encounter on a normal basis. I'm not exactly sure how the ions are emitted from the dryer, or what kind they are - which may provide more information.