Changing a server icon on Discord will not affect the icon shown by Ripcord, instead will continue using the same one that was used when first joining the server.

This is a known problem. You can force them to update by quitting Ripcord and deleting the discord_avatars.ripdb file (note that user avatars already update correctly as needed.)

The reason I haven't gotten around to implementing this already is that servers which change their icon frequently annoy me.

Noticed this when people were referencing the icon and I had no idea what they were talking about. Apparently it'd been like that for months. Perhaps a good default would be to update on restart? That should happen infrequently enough not to be a hassle.