Orthodox
Why do the Orthodox reject papal infallibility?
The Orthodox affirm that the Church as a whole cannot fall into error, but reject that this charism is exercised through one bishop apart from the whole college; Catholics reply that papal infallibility is narrowly defined and always exercised in communion with, not apart from, the universal faith.
Catholic teaching
Vatican I (1870) defined that the Pope, when he speaks ex cathedra as universal shepherd defining a doctrine of faith or morals to be held by the whole Church, is preserved from error by the same divine assistance promised to the whole Church — a rare and carefully limited exercise, not personal inspiration or ordinary teaching.
Biblical foundation
Christ prays specifically for Peter, 'that your faith fail not,' and charges him to 'strengthen your brethren' (Luke 22:32), and gives Peter alone the keys of the kingdom with binding and loosing authority (Matt 16:18-19), a personal commission distinguishing his role within the apostolic college.
Scripture references
Historical evidence
Long before the schism, Pope St. Leo the Great's Tome was read at the Council of Chalcedon (451) and the assembled bishops, including many Eastern ones, exclaimed 'Peter has spoken through Leo,' recognizing a unique doctrinal authority attached to the Roman see.
The Church Fathers
“Peter has spoken through Leo... this is the faith of the fathers, this is the faith of the apostles.”