From what I've read re the Inquistion, the Church per se never burned anyone. The Church, having no legal authority to put people to death, would hold a (twisted) "trial" to convict the person of heresy, then hand them over to the secular authorities for execution. The Church would not protect so-called "heretics" from the horrific secular "justice" of the times, but it did protect the faithful.

Just like in Jesus' day -- the Temple authorities tried Jesus, but handed him over to the Romans to be put to death.