According to worldofwar.net there will be no open Beta for Burning Crusade. They posted a quote where someone said:
„While we do plan to host a closed beta at some point in the near future, at this time we do not have plans for an open beta, like what was experienced just before we launched classic World of Warcraft. It’s always possible that our decision on this matter may change, so your best bet is to periodically check the website for updates, but I’d implore you not to get your hopes up, as it is very doubtful that these plans will change.“
Maybe Burning Crusade is just too bad and they do not want us to realize about it in advance. At least a former colleague who is still working at Blizzard Europe said he does not even play the Alpha at the moment, since there are many new things but nothing really breathtaking.