I'm pretty sure I've heard both of the following multiple times:
Transfinite induction requires the axiom of choice. False, though many applications of transfinite induction require axiom of choice (either in the form of the well-ordering theorem, or directly (though using transfinite induction together with choice directly is essentially the same as just using Zorn's Lemma)).
Transfinite induction requires the axiom of foundation. I guess some people get transfinite induction mixed up with epsilon-induction?