The strongest objection to the principle is about the canon. The Bible does not contain a table of contents. The list of books was recognised over centuries by the church, disputed at the edges into the fourth century and beyond, and settled differently in different communions. If Scripture is the sole final authority, something outside Scripture had to determine what Scripture is.
Protestant answers vary: that the canonical books are self-authenticating and were recognised rather than conferred; that the church's recognition was a fallible historical act that happened to be correct; that the question, while awkward, does not restore to the church a general authority to define doctrine. Catholic and Orthodox theologians regard none of these as adequate and take the canon as the clearest demonstration that Scripture emerged from within a living tradition and cannot be set over against it.
The Catholic position, stated at Trent and refined at the Second Vatican Council, is that Scripture and tradition flow from a single source and are interpreted authoritatively by the church's teaching office, which is not above the word of God but serves it. Eastern Orthodoxy speaks of Scripture as the pre-eminent part of a wider tradition that includes the councils, the liturgy, the fathers and the icons, and locates interpretation in the consciousness of the whole church rather than in a defining office.
The canons themselves differ. Catholic Bibles include seven books Protestants set aside at the Reformation as apocryphal, following the Hebrew rather than the Greek collection; Orthodox canons are wider still and not uniform among themselves; the Ethiopian Orthodox canon is the widest of any Christian tradition. A reader who wants to see the practical consequence can note that the passage most often cited for prayer for the dead sits in Second Maccabees, inside some of those canons and outside others.