The Heigan Dance was a term coined in patch 1.11, at the release of level 60 Naxxramas (thus refuting any claim it's a reference to the achievement - a system not in place until 3.0.2). It is in reference to Heigan's Eruption ability, which players must move around to avoid.
Three quarters of the room, not including the raised platform, will erupt for 6-7k damage (and standing in the middle of a quarter will deal the damage several times over, surely killing a player), so a player must 'dance' between different safe quarters. Simple. :)