For example, can I use David Cameron's face in an app, without permission?
You could run into these problems:
Copyright in the picture: you would need to own the copyright (e.g. by taking the photo yourself), have permission of the copyright holder or use a photo in the public domain.
Trademark in the face: a sufficiently famous person can have trademark in their appearance. If so, you cannot use it in such a way that a person could believe that that person endorses your app.
Model release some jurisdictions require a model release before a photograph of a person can be used for a commercial purpose. Search photo on this site for several questions on this.