The DMCA is pretty clear on this. If someone posts something to an internet service, and they get a notice, and then take it down, it’s no harm, no foul.
This appears to be YouTube at a glance, but Viacom is now claiming, “that YouTube is guilty of public performance of such content due to the way it presents and plays the videos“, because it allows for embedding and sharing (actually referrals) to the material.
This is going to get very, very ugly.