Have you tried this yet? I’ve gone round and round with Microsoft over this but I guess that it’s just not possible. If you have an App Part that is trying to read from a list inside your App using REST, it will not allow this as an anonymous user. Has anyone come up with a solution for this problem or am I not seeing things correctly?
Have you taken a look at http://social.msdn.microsoft.com/Forums/sharepoint/en-US/7387b9d4-56b8-4ede-b4fb-5868b81bd02a/access-denied-app-part-on-anonymous-public-facing-site?