While copyrighted material can certainly be found on YouTube, winning a case against them for copyright infringement is much harder than it looks. YouTube is protected by the Digital Millennium Copyright Act which, among other things, protects sites like YouTube from the actions of its users. Copyright holders are left with filing a notice with YouTube to pull their content down (which they will do), or trying to prove that YouTube moved outside of the safe harbor under the Act.
Viacom and other are trying to do exactly that, and are playing a $1 billion game of poker with Google, YouTube's parent company. But to add Apple to the mix, who simply show YouTube content but do not host it on their servers, adds a whole additional layer of legal complexity to the case. And it drags another war-ready litigation team to your front door. My guess is all of these hurdles will protect Apple, and we won't be seeing any litigation over their integration of YouTube into the iPhone any time soon.