Actual live auto-tuning has not made its way to Broadway (yet).
But as Jorge says above, several musicals use what is referred to as "sweeteners" to beef up large ensemble numbers. Basically pre-recorded ensemble singing that plays underneath the live singing. These are used either because of moments of intense choreography where they don't want the vocal power to dip, or because the show has a small ensemble and they want to increase the vocal output a portion of a song.
Additionally, we've seen plenty of shows use live sound effects on top of vocals. This is done in specific moments to give a vocal a stylized aesthetic, add reverb, etc. While these effects alter the ultimate sound of the singer, it isn't auto-tuning or pitch correcting them.
We've never had a Broadway show incorporate live auto-tuning. I'm sure there are certain producers who want to introduce it, but you can bet it will be QUITE the topic of debate once someone tries to do it.