WorldofWar.net is claiming via an unlinked quote that Blizzard will not hold an Open Beta for The Burning Crusade:

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.