{Hmmm.} I have not heard of this exact issue before, but it is very similar to a lot of the frame rate drop issues that people have due to hard drive read/write. While they report the bug to the dev team, you can try this as a work-around (and if it works, they have more information about what may be causing the issue).

Create a temporary folder your desktop. Open your documents folder, then your My Games folder, then Final Fantasy XIV. Move everything from there into temporary folder on your desktop. If you have a lot of custom configurations (chat filters, moved widgets around, macros, etc.) and do not want to re-make them, then move your "users" folder back into the Final Fantasy XIV folder. If you do not mind re-doing all your settings, then I suggest leaving the user folder out as well. Run the Final Fantasy XIV Config program; adjust your settings as you see fit. Run the game.

See if that helps.