I have never used a screen protector on my Note(or any other phone for that matter), so can't say with any certainty if it would cause the issue you're having. However, it wouldn't surprise me to learn that it would indeed cause such issues with the s-pen.
Removing the screen protector and seeing if the issue goes away would be your only real way of finding out for sure. It may also be a case of the particular screen protector you are using causing the issue, as having just done a Google search, it appears this can affect the s-pen.
The Note uses Gorilla glass 2 and shouldn't scratch easily anyway, so the need for a screen protector may be somewhat of a moot point. My note is protected from every day scrapes and knocks with a Samsung flip case and the screen has no blemishes on it at all after months of continual use.