For a client we wrote some code that programatically consumes RSS feeds and stores key information into a SQL data warehouse. When we decided to start surface SharePoint lists via RSS feeds, we ran into a small security problem. The RSS feed was requiriing an authentication. In order to only expose read access to this list anonymously, I had to go through several steps.
In central administration, you need to allow the ability to set anonymous access in the authentication provider as show below:

Make sure you pick your web application and then select your authentication provider:

Select Enable Anonymous access. This doesn’t automatically open your existing sites to anonymous access - it allows you to be able to set what will have anonymous access.

Within your site, you need to control how anonymous access will be applied. In Site Settings, select Advanced Permissions. Within Advanced Permisssions, select Anonymous Access as show below:

At this time you can control how you want anonymous access within your sites. For the purposes of this exercise, I only wanted to expose a single list to anonymous view access. So select Lists and libraries as shown below:

Now I have the ability to set anonymous access at the list level. Within the list I want to expose, go to List Settings and select Permissions for this list. From there select Edit Permissions in order to break the permissions for this list from the parent so they are no longer inherited:

After breaking inheritance on the permissions, you will have the ability to set anonymous access on the list. Under Settings select Anonymous Access.

To enable anonymous read access, check the View Items and click on OK.

The RSS feed for this list is now accessible anonymously.