*YEP, SPOILERS AHEAD*
My favorite opening was FF4's. It had so much things happening, and excellent music to back it up. First the airships flying, Cecil remembering about what they did at Mysidia and the whole crew regretting that. Then, for trying to state the crew's opinion, Cecil gets demoted from his ranks as the Captain of the Red Wings. He has a discussion about this with Cid and Rosa - a more emotional one with Rosa. Then Cecil and Kain leave on a journey with the great FF main theme playing.
FF9's opening is also great, IMHO. Garnet's memories along with beautiful FMV and peaceful orchestral music. That was followed by lots of unique things and mini-games that kept the beginning interesting.
I agree with what was said about the remakes having well done intros. FF1's opening is pretty cool even in NES. First the heroes save the princess, and then their adventure actually begins (with FF main theme which is really a great song). BTW this happened pretty much with Super Mario RPG too: first trying to save the princess and beating up Koopa, and then the game really begins. FF2's opening wasn't that good until the PSX version. It's not my favorite even in PSX, but an action-based intro FMV, a chase sequence and meeting with the rebel army kept me quite interested. FF3 is decent, although the FF main theme was once again a good plus.
I like FF5's opening because it's quite beautiful and the music suits the events perfectly. I always get chills when strings play, the camera zooms at Tycoon Castle and Lenna says, "The wind stopped! Has anything happened to father?", and then the king walks into the Wind Shrine only to get the crystal blown to his face. FF6's opening is excellent thanks to its music playing with the events. It's been a while since I played FF6 so I can't remember it too well, though.
FF7's intro FMV isn't that good by itself, but it's cool to get straight into action. FF8's intro FMV is awesome in both visual and audio departments. A problem is that the following events are rather boring, and the great stuff don't come back until the invasion of Dollet (about 30 minutes after the game's beginning).
FFX's intro is beautiful with Zanarkand music playing, but I'm getting tired of seeing that opening so many times. (It comes if you let the menu stay on, when you start a new game and when you finally (FFX SPOILER)reach Zanarkand, and you can only skip the one with the credits playing.) The first things are quite boring, then come the Blitzball match and Sin's attack. Those parts are cool, but after Tidus gets transferred into the other world, things get boring once again. For FFX-2, I just know it begins (FFX-2 OPENING SPOILERS)with a concert where Yuna is singing. There Rikku and Paine are introduced, and they're beating up some security guys. They also believe that the Yuna singing at the stage is a fake. However, I don't know how it continues after that.