How is it that Tommy Lee Jones plays Two-Face in Batman Forever, when Dent is black in Batman?
In Batman (1989), Billy Dee Williams appears as a Harvey Dent before his disfigurement, vowing to reduce crime by locking up mob boss Carl Grissom. Williams was set to reprise the character as Two-Face in the sequels, but the Harvey Dent character was deleted from the script for the sequel Batman Returns (1992), and Batman Forever (1995) director Joel Schumacher decided to cast Tommy Lee Jones as Two-Face.
Since Dent was white in the comics, and since Billy Dee Williams' role in the first film was rather small, and since the role of Batman himself had been recast anyway, it seems Schumacher didn't feel the need to maintain this particular continuity. (Joel Schumacher also shot down the prospect of a black Robin: After replacing Burton to give the film a 'lighter' tone, Schumacher recast Tim Burton's original choices which also included Marlon Wayans as Robin and Renee Rousso as the love interest.) And according to this site, Schumacher wanted Jones for the role from the very beginning:
"I always wanted Tommy Lee Jones," explains the director. "I didn't consider Billy Dee Williams for the role, because I think that he is a hero. I always see him like Clark Gable. I had just finished working with Tommy Lee Jones on The Client, and I thought he would be a great Harvey Dent\Two-Face."
In any case, changing a character's race is actually more common than you might think; you can find some rather blatant examples here. (cough 'Last Airbender' cough)
Artistic freedom allows him to do this. Besides, those early superhero worlds were 100% white. It may have been considered tasteless to have a black character play a villain, which could have been construed as racist by all those happy race-baiters.
If you're saying Dent appeared in an earlier Batman movie (I don't really recall those Burton ones too well...), then just remember that 3 different people played the part of Batman.