I never noticed Angel’s theme changing, but Buffy’s got rerecorded for the third (and subsequent) seasons. They also changed the font the logo was written in at some point, I don’t recall if it was for season 3 or earlier.
It was mentioned that Quantum Leap would change the scenes in the opening, but for the last season they also changed the theme. There was a lot of complaining about it, but I didn’t think it was bad. The final episode used the old theme, and for syndicated reruns, so did the rest of the season, at least I never saw the other theme again after NBC.
Someone already brought up The Drew Carey Show’s three main themes, but the last two seasons used a rotating set of remixes of each of the three. I don’t know how many different ones there were, but there were quite a few.
For the last season Alias had a different opening. IIRC, they actually showed the actors, not just names on a black background.
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The Office** recently added Andy to the opening, in addition to Michael, Jim, Pam, Dwight, and Ryan. There was also an alternate opening for the Superbowl episode with the entire cast actually getting shown, and an alternate Michael Scott Paper Company used for one episode.
Already been posted that The Simpsons redid the opening when they moved to HD, but they also reanimated it once before, I just can’t recall if it was for season 2 or season 3. Of course, they have many shorter versions used depending on how long the eps run, and have had a few alternate versions along the way (a live action one, one with everything looking wintery, etc)
Similarly, One Tree Hill used to occasionally drop the credits for “special” episodes (my guess is they just ran a little long and needed the extra time), but when they jumped forward 5 years they got rid of the opening for good. (And I hated that opening to Grey’s, :p)