The worst that's ever happened is a few years ago, Lucasfilm handed out some cease and desist orders regarding some fanfic he found distasteful...and to be honest, I think those were actually people who posted on the net
As for Star Trek, it has historically been a fan-friendly fandom. Roddenberry knew what side his bread was buttered on, since the fans saved his show not once but twice. He and TPTB who succeeded him were very open to fanfic, because they understood how much interest and enthusiasm it helped to generate. They even set up writer workshops and contests to encourage fans to break into episode writing and profic novels. Trek author Jean Lorrah started out as a fanfic author and went pro. She still writes fanfic. I see her every year at Media West. She'll even autograph your novel for you at her dealer table.
Honestly, as long as you don't make a profit and don't do anything to tick off the creators, you're generally safe whether you have your work published in a print zine or on the web. My main worry is that once legislation catches up with technology, I think it's going to be net-based fanfic that gets targeted, because it's much more easily accessible than zine work. Fan fic and fan art is not a phenomenon unique to the internet, it just adapted into the internet.
Any way, if you've ever been to a con in the last 30 years, I know you've seen zines in the dealers rooms. It ain't nothng new under the sun.
