I don't think anyone is suggestioning that the paper will have an impact, but it is part of the overall first impression. If you send it on nicer paper, it helps put out a nice first impression. If you send it on plain white printer paper with faded type, that is going to send out a bad impression. Again, it's not going to be the deciding factor (if it was, how the hell would want that job anyway?), but I'd think, for some anyway, it probably plays a part, even subconsciously.
Of course, if what is written on the paper isn't good enough, well then who cares.
I had an interview about in which the hiring editor told me that I was the only person that did not have an error on the resume or cover letter. This was for a copy editing job.