What is the proper role of the Thirty-nine Articles today?
Well, I am tempted to answer that question by saying "more than the Anglo-Papalists desire, and less than the Evangelicals want!"
Anglo-Papalists would really like to forget about the Articles of Religion altogether and shut them off into the "historical documents" category, if not loose them altogether. They recognize, quite rightly, that they are a road block to remaking Anglicanism in the image of 1930s, 1950s or modern Roman Catholicism.
Evangelicals, on the other hand, want to make the Articles into a narrow and binding Confession of Faith. This has a bit more justification behind it than the Anglo-Papalist position, but it still has its problems:
Firstly, Anglicans have never regarded the Articles as a Confession of Faith in the narrow sense, but rather as a broad affirmation to the Biblical version of Christianity. We have been required to subscribe to the Articles as "containing nothing contrary to Scripture" rather than asked to bind ourselves to a particular version of Biblical theology. This is a fair, logical, and Evangelical way of making one's subscription, as it commits us not to the personal opinions of a group of sixteenth century theologians, but to the doctrine of Scripture.
Secondly, I would also like to point out that the Articles have never been a stand-alone document. Not only do they refer not just to the Scriptures, but to the Early Fathers (specifically Jerome), and to the Book of Homilies, but we have always been bidden - for example, by Abp. Matthew Parker - to interpret them in the most catholic sense.
Thirdly, theology did not stop in 1563. The development of culture and society in history throws up new challenges to orthodox Christianity from time to time. The Articles do not answer these questions, but they do give us a theological method with which to approach new challenges. This method begins with Scripture, and looks to the Early Fathers and Councils of the Church to guide us as to the authentic teaching of Scripture. Private opinion has little or no place in our tradition.
Anglicans are first and foremost "Bible Catholics." Unlike the Roman Catholic Church, we teach that all doctrine necessary to Salvation is to be found in the Canonical Scriptures. We also take a distinctly Pauline and Augustinian approach to the doctrines of Justification and Sanctification. However, we also embrace the Catholic tradition of the Church that is rooted in the Fathers and the Councils in so far as it is compatible with Scripture.
The Articles of Religion are therefore part of a wider theological tradition built on the Bible, that includes the ancient creeds, the Early Fathers, and the Ecumenical Councils of the Undivided Church. They are not a stand alone document. Anglican tradition also affirms the principle of "adiaphora" and also affirms the need for dignity and beauty in worship, following traditional Catholic Uses in so far as they are not contrary to Scripture. Anglicans should reject all Papal additions and Puritan subtractions from the Faith of the (Early) Church. Unfortunately there are enthusiasts on both sides who will not be content unless we embrace the errors of Rome - or for that matter Geneva. As we are bound by truth and not expediency, we cannot in good conscience do this, but must remain faithful to the Bible, the Fathers and Ancient Councils, and our liturgical tradition as it is enshrined in the historic BCP!