My company participates in a number of job fairs during the year, and I often draw the short straw and have to review the big stack of resumes that come back from these.
Until recently I would have answered 'just use copy paper, nobody in the technical world cares'. And that's probably true most of the time.
I was recently given a resume stack from a job fair, and I notice right in the middle of the stack there was a resume printed on thick, ultra-white paper. It really stood out, so I pulled that one out first.
It was the resume of a graphics designer. She had some colorful, fun, and interesting graphics and the whole presentation was just beautiful. I took that resume straight to our UX guy, who now uses this person as a contract artist. In this case anyway, the paper she used got my attention.
Would it have mattered as much if she was applying to be a Java programer? Probably not, but it wouldn't have hurt her either.
If I ever go looking for a gig at a job fair I'm going to use a better-than-copier-grade paper for my resumes. I'm not capable of colorful or fun graphics, so I won't have that.
Once you are in the door for a physical interview it's less important, but couldn't hurt to use nice paper.