Quite possibly, if the following quote from Matt Smith is any indication.
“It won’t just be a televised event, I think it will be a live event. The way it’s transmitted on TV will hopefully be an inventive thing – something different.“
This is by no means an official confirmation, nor was Matt necessarily talking about an actual live episode, but it’s certainly something interesting to ponder. We keep hearing rumours for the 50 Anniversary episode, from a multi-Doctor episode to old companions coming back, but have yet to have any of them confirmed.
Ultimately, I don’t think a fully live episode is likely. Doctor Who incorporates many digital effects which are only possible in post-production and it’d be impossible to do live. Of course, this doesn’t rule out live segments within the episode itself, which would be an interesting way of delivering the scenes which don’t involve effects.
Perhaps if they had some kind of giant secret they were going to reveal. Perhaps a question they were going to answer. A really old question. One hidden in plain sight all these years…