I recall a Lutheran theologian writing an article in a Lutheran publication arguing intelligently that the concept of Sola Scriptura preceded Luther in the late medieval/early modern Catholic Church, with Erasmus and others holding to this school of thought in Catholic Germany. So this idea itself wouldn't have been a problem enough for the Catholic Church to dogmatically reject it at the time.
That said, Orthodoxy doesn't have this idea, and there's a consensus that it conflicts with Orthodoxy, with its reliance on Tradition. The most official formulations of Sola Scriptura seem to be that the Bible alone establishes all dogmas/articles of faith (source: Luther's Smalcald Articles) and is the sole judge of every teaching (source: Formula of Concord). Luther used the explanation that everyone checked their teachings by the Bible, and that therefore the Bible ultimately was the judge of all teachings.
One tricky part of understanding and applying the Sola Scriptura concept is that it involves polemical writings that can have exaggerations and leaps of logic. So first: what is an "article of faith"? The most natural meaning would seem to be that it's a credal theological statement, like the Nicene Creed. But Luther in his polemics (eg. in the Smalcald Articles) complained that the Catholic Church at large and Henry VIII's Catholic Church in England in particular were turning issues like relics and ritual vessels into "articles of faith", by which he seemed to mean mandatory teachings. According to Luther, the Bible did not specify types of ritual vessels and some other ceremonial specifics, and so the Catholic Church should not mandate them. However, on critical reflection, Luther's line of reasoning raises a line of issues.
- First, the Catholic Church might not agree that it considered its mandatory rules, like what vessels to use, to be "articles of faith." Luther was using an argument that by mandating the ceremonial vessels, the Church was turning them into faith articles, but the Catholic Church might not agree that just because something is mandatory that it counts as an article of faith.
- Second, the Lutheran Church for practical reasons ended up making rules on some of the same kinds of things. So a Lutheran diocese can have a policy that congregations are supposed to use cups for giving communion. Luther's polemical argument might claim that such Lutheran policies are not "mandatory", but nonetheless in real life practice, the Lutheran church is going to treat them as requirements. Otherwise it's easier to end up with Clown masses. The Lutheran Church might respond that they are enforcing the Luhteran policies for practical reasons, not because they are Lutheran faith articles. But in that case we end up with the substance of Luther's argument in the first place - Luther complained that the Catholic Church, by making rules on communion was de facto treating the rules as articles of faith. And in this instance, the same kind of argument would follow from Luther's logic.
- Third, another issue is how much and what topics the Bible actually covers. Does the Bible speak (A) just on salvation issues, or (B) on necessary faith articles, or (C) does it literally cover every single doctrinal topic that might arise? For the Bible to be the literal sole judge of every doctrine as the Formula of Concord asserts, it would seem that (C) the Bible would have to actually speak on every issue. However, according to Luther there were actually topics that Christians debated that the Bible did not cover, like the issue of ceremonial vessels.
- Fourth if one says that the Bible covers an issue (A, B or C above), them must it do so directly and specifically, or may it just do so in a very indirect way? The Bible never specifies a stance on infant baptism in particular, so Protestants typically try to cobble a position together indirectly from circumstantially relevant Biblical statements, like a case when a whole family of unknown age was baptised. And if we can say that the Bible gives a position indirectly on infant baptism, and we can take "Biblical" positions to include implicit, circumstantial ones, then it seems that we could assert that the Bible gives a position indirectly on communion vessels too, as Jesus "took the cup," etc. But in denying that the Bible takes a position on communion vessels, Luther seems to exclude relying on indirect statements by the Bible.
Luther was raising an issue of procedure- how does one go about establishing and judging a doctrine.
- Orthodox typically, and technically even many Protestants including Luther, do not strictly follow a rule that literally the Bible alone judges every teaching. These theologians don't literally quote the Bible alone. Of course, Luther himself added that one could use other writings, and then he would assert that it was actually the Bible making the teaching, not those other writings, as the Bible was the judge of all writings.
- Offhand I can't think of a dogmatic credal statement in Orthodoxy that can't be asserted to be at least indirectly implied by some part of the Bible. And perhaps that's true for Catholicism as well. In other words, as a matter of proecedure, I can't think of a case where one could not use the Bible to at least indirectly establish what EOs or Catholics openly would agree to be an "article of faith." Take for instance the Nicene Creed. The Creed is an article of faith and Orthodox and Protestants would agree that its theological substance can be established using the Bible.
- On the other hand, I think that theoretically Jesus and the apostles could have given a theological teaching that they didn't specifically mention in the Bible. And in that case, the procedural side of Orthodoxy would allow for those "nonBiblical" apostolic teachings. IMO there are actually a couple cases like that, like Infant Baptism, where the apostles and Tradition took a position but the Bible didn't mention it.
- Then there's the issue that Orthodox theologians would say that the Bible itself does not actually teach Sola Scriptura- the Bible nowhere specifies that the Bible alone makes every faith teaching and judges every teaching. At best, an apologist for Sola Scriptura would have to look to statements in the Bible emphasizing the importance of Scripture and then conclude that this indirectly implies that the Bible is the only establisher and judge of these things.