The first trick for RSS feed web part is you should change authentication provider settings from NTLM to Kerberos with AD SPN setup if you need to display any SharePoint list as RSS feed. You do NOT need to setup Kerberos if you just need to display external public RSS feed such as http://www.microsite.reuters.com/rss/businessNews. The way to setup is straight forward and have not changed since SharePoint 2007. You could check from central admin on each webapp as described here. I noticed if you do not setup AD SPN, the webpart is still not working. Here are the procedures to check whether Kerberosis in deed working for the webapp.
Check SPN setup using command setspn -l domain\installaccount. Here is the example output from the command. You should see three entries on the server and fully qualified name.
C:\>setspn -l na\installaccount
Registered ServicePrincipalNames for CN=,OU=CORP,OU=Service Accounts,DC=domian,DC=company,DC=com:
In order to confirm the user is indeed login using Kerberos, you could open the Security logs and filter for “Event ID: 4624”. You will see Kerberos as Authentication Package. More details in the future if you are interested in this topic.
The second trick is you only need to setup Kerberos on the site that is using RSS feed webpart. The displayed site does not need to be Kerberos enabled. I have the screen shot below that has three RSS feed web parts. As long as the web part site is Kerberos enabled, it can display both Kerberos enabled or NTLM enabled site lists.
The third trick or issue is I'm not able to setup to auto refresh the web part with the latest feeds. It is still same as 2007 that it only refresh after IISRESET! I'm able to find the AJAX options and setup to enable Asynchronous Automatic Refresh and Update as in the below screen shot. However, the web part displayed error "An unexpected error occured while rendering the webpart.". There is even an typo in the error message!
As of today August 30, Microsoft has provided a workaround to refresh this webpart. Pleas refer to here for details on the workaround.
The fourth issue is I'm not able to display the feed across the farms. See previous third RSS feed web part. I'm getting error "The RSS webpart does not support authenticated feeds." that is same as you have not setup Kerberos.
There are some other issues you may encounter fro external RSS feeds with SSL and different sources. We have noticed people mentioned they could display feed across farm we will continue to debug.
Finally Microsoft confirmed this is not support to auto feed refresh to display SharePoint lists. However, we have identified a workaround that could refresh the webpart even it's not using Ajax. Please refer the workaround and let me know how it works.
If you have error "The RSS webpart does not support authenticated feeds." on SharePoint 2013 on-premises, you might need to sett the aspnet:AllowAnonymousImpersonation to false in web.config as Brandon mentioned. I would be cautious to change it before complete testing for other webparts especially Access apps, any BCS webparts, and custom webparts using security token.
If you try to display SharePoint online O365 RSS feed from on-premises even after you configure the trust, you will still get error like 'ProtocolError occurred trying to complete the request. The server returned a status code of : Forbidden and the status description is : "Forbidden"'. I will do more research how we can configure this.