In cases involving infringing postings at Google listservs, a copyright owner (such as the New York Times or Newsweek) would likely pursue the poster and Google.
Safe harbor. Google will likely be shielded from a lawsuit by claiming the "safe harbor" benefits of the Digital Millenium Copyright Act (DMCA). Section 512 of the DMCA allows an Internet Service Provider (ISP) such as Google to avoid liability for infringement provided that the ISP promptly removed the material upon request and provided that the ISP has met other requirements. In other words, if Newsweek or the Times request removal, Google will issue a take-down notice, and if the poster fights the notice -- for example, the poster claims a fair use defense -- the copyright owner can sue the poster in federal court and sort out the matter, there.
What about you? It's possible (though unlikely) that you, as manager of the listserv, could be drawn into the line of fire. In one case, a federal appeals court indicated that if a moderator (under the supervision of an ISP employee) reviews and approves all postings, the ISP could be liable. We doubt whether you'll be named in a lawsuit. Still, because you're aware of a potential problem, you should provide your copyright policy to all subscribers, notify the poster that only links should be posted, and seek guidance from Google for the issue if the poster doesn't correct behavior.