Whilst it could be seen as a different scenario, one that sticks out is Dark Knight's original Darkside Aura, first shown in the Hevensward job action trailer:
HW Job action trailer (Time stamped to Dark Knight)
As shown, they used to have dark/black flames around the feet which was removed on release. There might be other examples I am unaware of, however, this is one case where an animation has been changed before release. The aura was also large and obnoxious on release, so it was toned down and made a bit dimmer.
There are also sound effects that have been changed for various jobs, however since this is just sound effects and not visual effects, I won't elaborate further, just thought it would be something worth mentioning. I also haven't mentioned changes where animations were changed to match a new targeting scope, like Tornado Kick going from single target to AoE, or Overpower going from a cone to circle among others.