Is it really even news when an app essentially dedicated to copyright infringement is claimed to violate copyright laws? Popcorn Time isn't a general-purpose app that happened to be used for copyright-infringing purposes sometimes. It was designed to help people infringe copyrights by making it as easy as possible.
What's news is that this isn't how the DMCA is supposed to work. It's supposed to apply to copyrighted material. Not material which when downloaded and run will connect to a P2P network and attempt to download material, which has some probability of being infringing.
This case actually seems relatively flimsy; and as the DMCA notice admits, it hinges on a previous ruling that
“the distribution of a product can itself give rise to liability where evidence shows that the distributor intended and encouraged the product to be used to infringe”
Unfortunately there's no way this notice will be challenged in court, so we probably won't find out if it was legally valid or not.
If the DMCA doesn't apply, then the MPAA could immediately sue whoever is hosting these files for contributory copyright infringement.
The DMCA is a shield, not a weapon. Essentially, you are shielded until you get a notice of the infringing content.
But there is no legal force behind a DMCA. If you get one you can tell the company to fuck off. But you are now liable for any infringement, if proven in court.