It seems that no one has any solution for me :(. But, after messing around with it, I have found an explanation and a solution for it and I think I should share this with you. Sorry for the lateness in the response, though.
Here is the story.
As you probably already know, after supplying the credentials, Facebook returns the access token in the URI which is followed by the # symbol. Now, the BlackBerry Facebook API overrides the handleNavigation() method to ensure that, if the URL contains the access token, it will not be processed. This used to work in earlier versions (prior to BB 7) but somehow, it seems that URL redirection in BrowserField are no longer calling handleNavigation(), but rather handleResource() directly (I don't know why and, to my knowledge, there is no documentation for this change anywhere). The result? It always tries to process the URL that contains the #access_token in it.
Worst, the BB BrowserField thinks that # means a jump to a portion of the page (which doesn't exist). Other browsers (including the BlackBerry Browser) seems to understand that already and print only "Success" on the screen, while the BrowserField does not. And when that happened, the application just stopped working and crashed, sending me back to the home screen.
So what I did was I changed the source code of the BlackBerry Facebook API so that handleResource() now checks by itself whether the access_token is in the URL. Then, the application works fine.