It's hard to answer without knowing how the product exactly works, but I can tell you I see two main legal issues here already.
1) Copyright issue related to the content of the videos. The classic problem of user generated content: what if somebody uploads the leaked Jay-Z video or upload something he created but with a non-original soundtrack without obtaining clearance / pay synch royalties.
Luckily, this problem is relatively easy to solve thanks to the DMCA safe harbor provisions. You can google that for more info on how to do it, but in short if you set up a takedown process for infringing material, you can escape liability.
2) Secure the right to source/republish content from other websites. Unfortunately, the only way to solve this is going to the TOS of each one of them and check if you have the right to do that. I think YouTube allows it, but there might be limitations.
I definitely recommend you check with your lawyer before starting this venture. I don't think he'll say "this is illegal"; he will probably blurb a complicated explanation that would not provide you with a clear answer. It is up to you, the entrepreneur, to ponder the legal information, assess the risks, and decide if it's worth or not. "Legal risk" is never zero.
{insert usual "this is not legal advice" disclaimer here}
