On my MacBook Pro, my RSS feed reader of choice is ReadKit, and I like it a lot, because of its UI similarities with Reeder for Mac, and also because of how it handles my Pinboard bookmarks (as if they were feeds, in fact). 

Like many other feed readers, ReadKit displays the site’s favicon next to each feed:

Sometimes, though, favicons may get corrupted or the site owner may update them (as I myself did with my site). It’s just a minor detail, but in such case I want the feed reader to show me the updated favicons. All feed readers usually keep a favicon cache to avoid loading them every time you launch the application. It usually resides in a folder inside [your username]/Library/Application Support/[app name]. Typically, to force the feed reader to reload all favicons, you’d delete such cache, but with ReadKit I couldn’t find anything of the sort. ReadKit doesn’t even create a preferences file in the usual [your username]/Library/Preferences folder. 

After a deeper search with the always excellent Find Any File, I found out that ReadKit stores favicons in a favicon folder inside [your username]/Library/Containers/com.webinhq.ReadKit /Data/Library/Application Support/ReadKit/ Simply delete the contents of that folder (keeping ReadKit closed), and relaunch ReadKit. All favicons will progressively reload.