At Joan's trial, Bishop Cauchon tells everybody that, "The mighty structure of Catholic Christendom […] may be […] brought to barbarous ruin and desolation, by this arch heresy […] Protestantism" (6.74). But what exactly is Protestantism and why is it so dangerous to the Church? Warwick gives a good definition, saying "It is the protest of the individual soul against the interference of priest or peer between the private man and his God" (4.117).
Basically, Protestants in this novel believe that they should be able to talk to God themselves rather than having a priest do it for them. You can probably understand why the priests might not be fans of this idea. If everybody goes around communing with God on their own, won't all the clergy lose their job? Cauchon and his peers feel this is very dangerous because your average everyday Joe doesn't know enough to interpret God's will. They think it takes a trained and educated Churchman to get it right. For example, how is a layman supposed to know the difference between God's voice and Satan's?
Joan, of course, thinks she has every right to commune with higher powers on her own. She claims to the very end that her voices are sent from God, despite the fact that the Church tells her that they're demonic in origin. Joan even goes so far as to say, "I know that your counsel is of the devil, and that mine is of God" (6.233). Whoa, that's a harsh accusation. Not only is she claiming their interpretation is wrong, but also that it's been influenced by the Devil.
What's ironic is that Joan doesn't start out as a rebel against the Church at all. When she first meets the Archbishop at Charles's court she is extremely reverent, so much so that the nobles make fun of her. Even for most of her trial she is generally respectful of the Church as an institution. Nevertheless, her faith in her own interpretation of her voices over the Church's brands her as a heretic. The Church sends her to the flames because they can't allow this symbol of Protestantism to survive.
Note: Once again we have an anachronism. The word "Protestant" didn't come into popular use until Martin Luther's Protestant Reformation in the 1500s. Shaw, of course, is well aware of this, and chose to include the term its modern implications.