No. The Mass File Download app fully respects Salesforce security and does not override permissions. It adheres to file-level, object-level, and sharing/access settings of the current user. If a user has the “Query All Files” permission, they can see all files and attachments. Otherwise, they will only be able to see the files that are already shared with them or those they have access to. The app simply provides a UI within Salesforce for selecting and downloading files, without changing or bypassing security rules.