In Enterprise, the credits at the beginning of episodes with the mirror universe are different. Did any other Star Trek series have altered introduction scenes during mirrored universe episodes, or for other special kinds of episodes?
Of the 4 Star Trek series that include episodes in the mirror universe, Enterprise is the only one to include a unique intro related to the mirror universe.
As for more general changes, there are no other intros that had significant thematic changes. As pointed out in the comment above, some episodes had minor differences like the direction the credits moved, but nothing very noteworthy.