Funny you should bring this up....
The application to change numbers only mattered if he'd stayed with Cleveland. Players have to apply in advance to the league to change numbers if they stay with the same team, but if they change teams they can pick any number they like. As long as it isn't retired, or already taken. There is a league rule that if a player changes teams and wants to keep his old number, and it's already taken, the player who has been in the league the longest keeps the number if he wants, and the other player changes. This doesn't apply if a player changes teams and wants to change numbers - if it's already taken, tough.
Now, you are probably wondering why I went into this long-winded explanation about NBA rules for jersey numbers? Well, it's because not only can't LBJ take the retired #23 in Miami, but he can't change to #6 either. Even though they have only one player under contract at the moment - Mario Chalmers - guess what number he wears??? :)