Just because you didn't see any microphones doesn't mean there weren't any. I suppose it didn't occur to you that they obviously weren't lip-synching all the dialogue too and that certainly wasn't hitting the back wall of the theatre on the merits of projection alone.
So even though you've explained why you think there wouldn't have been lip-synching in Tarzan
for fairly sensible reasons above, you're going to believe the actors were indeed lip-synching because of an unsubstantiated comment, even when someone else has offered the opposite information from an apparently first person experience of 3 performances of the show? Interesting application of logic there...
Well, that certainly depends on the musical in question and certainly the mix of abilities in terms of proficiency in acting, singing and dancing respectively varies from performer to performer. Furthermore, extraordinary production challenges might require a tailor-made solutions. There are no absolute rules, although there certainly are conventions that a generally followed (as they should be).