From the message "Facebook API Error: This IP can't make requests for that application. ", it definitely sounds like a Facebook issue. For some reason, they don't like your IP address. Even if the IP is new to you, it's most definitely been used for other sites before, and likely been blocked there.
The Linter doesn't check the IP address the site is on because that's just scraping the front-end contents of your site. Likewise, the FB Developer Area will let you setup an application, but it doesn't necessary know what IP that domain will be run from.
From the information you've given me, and the error your seeing, it seems like your IP address, as the error mentions, can't be used to configure the settings for the Application. JFBConnect relies on the API to make requests from your site to be able to set the configuration details we recommend.
If you can't run the Autotune tool to configure you're app, I'd recommend setting up the app manually in the Developer area with the settings we recommend in the Autotune area. You won't be able to 'see' those settings reflected in the Autotune tool, but that shouldn't affect your app. Once configured, try authenticating on the front end (or do other Facebook stuff) and see if you continue to run into problems. That will help us narrow down how to proceed. If you continuously see that error about your IP though, it's indicative that the IP address is in fact blocked.
I hope that helps,
Alex