An important thing to keep in mind is that it was not the Church carrying out these punishments, it was the civil authorities. During the Middle Ages, a nation’s religious affinity was a core aspect of the nation’s identity. We see something of a parallel in Muslim nations today, with a significant distinction being that the Bible did not define national laws, whereas the Quran does.
When heresies popped up, they were considered a threat to national order, and were dealt with accordingly; much in the same way theft or murder is handled by the courts today. Even the ‘infamous’ Inquisitors operated through the offices of the national leadership, though also through the direct permission of the Church. Incidentally, rather than the blood-thirsty torturers and persecutors depicted in modern fables, the Inquisitors placed limits on practices which were common in the day, specifically, on the use of torture as a means of obtaining a confession. There are records of people who, when accused of a civil crime, would commit an act of blasphemy so that they would be tried by the Inquisitorial court instead of the secular court. This was because the inquisitors only used torture in very specific cases, and only within prescribed bounds, whereas the civil courts of the time saw torture as the primary means of interrogation, and did not generally place limitations on the type or duration of torture used.
So, were there burning in the name of Christendom? Certainly. Was is the Church doing them? Nope. It was the national authorities, who saw heresies as a threat to civil order, and a direct affront to the authority of the nation’s leadership.
As for the Church’s views on it, I’m not really aware of any specific statements outside of the belief that a civil authority has the right to exercise the death penalty when the situation calls for it. The question of its necessity or propriety in these instances is certainly up for debate, but one must always remember the social and political realities of a time when discussing history.
