This question is very similar to the one I had asked before: "When was it first said in Christianity by a Church Father that heretics must be put to death?", however, there is still a difference. In this question I am not looking at the approval of taking someone's life, but rather at the first known case of an approval (by any Church father) of inflicting physical tortures on heretics, perhaps, accompanied by cutting off some parts of their bodies (tongues, noses, hands, etc)
The Definition of Torture
The first thing to identify in such a question is the definition of torture. Torture is a fluid term and similar or even exact same actions may seem like torture against one person but not against another, so there is a subjective nature to the question.
There is also the consideration of certain execution methods actually being torture. As Christians, we are familiar with crucifixion, which involves a prolonged death and great suffering. During medieval times, drawing and quartering, flaying, and burning at the stake all are meant to kill, but are equally meant to torture. Other methods of execution that are today considered inhumane in many cultures would be stoning, hanging, and beating to death, however, these methods do not necessitate the prolonging of death, and if done in a particular way, can actually be quite swift. Also, until the banning of "cruel and unusual punishment" torture as we know it was typically just part of the penal system. It was viewed as necessary for either deterring crime or exacting God's punishments until roughly 1690, when England's bill of rights officially made torture illegal.
We are also going to neglect any torture that was not sanctioned by legal officials. The nature of this question necessitates that only examples of torture that public, Christian religious officials allowed or ordered. To further complicate things, we are looking for the earliest example of hypothetical approval of such actions. This proves more difficult to demonstrate unless specific torture events are mentioned.
So the definition we will work with is the WHO 1986 definition of Organized violence:
So whether an event is considered torturous or not has nothing to do with it ending in death. Instead, it must include:
Definition of Heresy
We must also define heresy. Heresy, like torture, has also proved to be a fluid term over the centuries. Originally, heresy first came into use when Christians found that some small groups, such as the Gnostics, held slightly different beliefs on key dogmas which made them incompatible with "the orthodoxy of Christianity."
Irenaeus is credited with thrusting the idea of Christian orthodoxy and safeguarding the religion from heretics in his work Adversus Haereses, which he wrote to discredit Gnosticism and encourage the traditional elements of the Church.
Generally, heresy has been defined as anything that is not in full agreement with "Christian Orthodoxy." What has changed often over the centuries is opinions on what Christian Orthodoxy is. So we are left not defining heresy ourselves, but leaving it up to the accusers. Whether we would call any person or groups heretics is irrelevant.
Where did it start
Torture has a long history thousands of years older than any Christian example. When looking closely at the New Testament it is difficult to make a case for torture, killing, war, or any violence. So how did the Christian church go from a persecuted and tortured group in the first and second centuries to an advocate and performer of some of the most heinous tortures?
It started with the Romans. Roman officials commonly tortured enemies and criminals, often with the intent to execute them afterward. And the old Roman Empire slowly became the Roman Catholic Church starting with the conversion of Constantine I. Irenaeus set the stage to confront heresy head first. Later, Augustine came to the conclusion that "compulsion" and violence were acceptable to turn the heretic from his ways, arguing that Paul was struck by Christ Himself before his conversion and became the most successful apostle. Augustine was accepting of fines, imprisonment, banishment, and moderate floggings, but likely would have been disgusted at the treatment of people in the Middle Ages.
But by 1252, Pope Innocent IV, himself, issued a papal bull that authorized the use torture against heretics, giving us the single most obvious approval and mandating of torture for heretics. This bull read:
Pinning it down to the first Christian approval of torture is difficult because it was a long road from Augustine to Innocent IV. There are quite a few examples of people being executed in horrific ways, such as burning, but whether religious officials approved is difficult to determine. Again, because our modern definitions of torture are different, some punishments given at the time were not considered torture, but just part of the punishment. Paschal I, who was Pope in the ninth century, is said to have blinded his enemies before they were executed. The same is said of Hadrian III. Whippings, beatings, branding, and removal of hands or legs were common punishments for many crimes, including heresy since Constantine I. Whether they were considered torture is almost entirely subjective, but most of them were approved and even ordered by religious officials.