API Can not download files under app Permission type App folder using API?

When using an access token for an "app folder" app, that token will only be able to access files inside the special app folder created specifically for the app. By default, this is at /Apps/<app folder name>. That app folder is the implied root for any paths referenced by an access token for an app folder app.

So for example, if a full Dropbox app access token makes an API call to access "/test.txt", the Dropbox API will look for something at "/test.txt" in the account.

If an app folder app access token makes an API call to access "/test.txt", the Dropbox API will look for something at "/Apps/<app folder name>/test.txt" in the account. In that case, if there's nothing at "/Apps/<app folder name>/test.txt" you'll get an error. 

By the way, to download files, make sure you're using /2/files/download (or the respective method in an SDK or library) to download files, and not /1/files, as the latter is part of API v1, which is deprecated and being retired soon.