Is this happening with one movie or does it happen to all of them?

Actually this happened with LOTR - Fellowship of the Ring, Crouching Tiger, and Amistad if I remember correctly. It didn't happen for The Last Emperor. Maybe it turns on the subtitles fully for movies where they have subtitles occasionally throughout the film?

Do you have subtitles when playing the movie on your computer? The prob could just be your dvd player.

Yes, the subtitles play on m y computer (PowerDVD) just like on the DVD Player. I think this has to do with the fact that the movies (like LOTR) have subtitles in them, even with subtitles turned off. For instance, even with subtitles turned off, they show up when the elves speak Elvish. Same with Amistad and Godfather. Any clue on how to turn subtitles automatically off when making 2 disc backups for these types of movies? Or if there's no way to do it with DVD XCopy, any other programs I can use to make two disc backups?

There is no option for subtitles as they aren't supposed to appear by default. Not sure why you have this prob. You can try a different prog or just turn them off with your remote.