Liability for copyright infringement can arise directly or vicariously.
A person can be direct liable if he personally makes an unauthorized copy of someone else's protected work. A third party can also be held liable for that infringement if he has knowledge of the infringing activity and makes a material contribution to the infringing activity.
If an app is basically only good for downloading YouTube videos, the vast majority of which are protected, a court will likely infer that the app was made for the purpose of facilitating copyright violations. That was basically what happened with Grokster, and the U.S. Supreme Court agreed that Grokster was liable for all the copyright infringement happening through its app: Metro-Goldwyn-Mayer Studios Inc. v. Grokster, Ltd., 545 U.S. 913 (2005).
The best way for the developer to get rid of the IP complaints is probably to disable the app.