Fancy paper can hurt your chances.
Everything you do or don't do says something. If it's in default font like Times New Roman, it says nothing. If you change the font to Comic Sans, Arial, Roboto, etc, that says something about you.
Where I live, people who put significant effort into marketing are covering up for poor quality. If I get a fancy paper, my first thought is "What's this guy covering up?"
It's an unfair generalization, but you associate yourself in the group of people who spend more time pretending to do work than the guys who are too lazy to do work. It's not an instant rejection but I'm going to keep an eye out for any signs of an office politician.
Another thing this says is that you're spending effort getting a job.
I have resumes of very impressive people - all of them are messy and ugly. They get poached and recruited without any effort on their end. I ask for their resume and their response is along the lines of "Sorry, my resume is 2 years outdated because the last few people hired me without looking at my resume..."
Exception is for UX/graphics guys, salespeople, and anyone with a business degree, as this is part of their job.
If you're joining a startup, this can be an extremely negative thing. Startups care about every cent you spend, and this can come up as being a little spendthrift, even if the paper doesn't cost that much.