Yes, but in a criminal case guilt must be proven beyond a reasonable doubt. And in this case it means that it must be proven that the owner of the forum is himself both the uploader and poster of the links.
Just a "admission" without more on a forum does not constitute proof.
DMCA 512h subpoena would easily show who uploaded the materials. Copyright infringement is rarely a "criminal" case unless you're talking a large scale operation. This would be a civil suit, and it would be part of the discovery phase to file a 512h subpoena with the filehost.
And what if the forum doesn't keep logs? There is no legal requirement to keep logs.
However, in order to qualify for the DMCA safe harbor, the service provider must have a registered DMCA agent. It requires filling out some paperwork, and paying a onetime fee.
And to the other poster, yes a forum may qualify for the DMCA safe harbor provided that it has a registered DMCA agent.
Whether links may infringe copyright is really an open question.
I seriously doubt Djboutit has went through the trouble of registering himself as a DMCA Agent. Unemployed wannabe DJs living at home rarely dot their 'I's or cross their 'T's. But his admissions/posts made on this forum as well as the many others (the ones he hasn't been already banned from that is) would help craft a beautiful narrative of a copyright infringer who thinks he's under the radar. It would be so easy to make an example out of him.