Let's say that some time in the mid-to-late 14th century, the Divine Comedy by Dante Alighieri is accepted by the Church to be the last book of the New Testament after the Book of Revelation. Dante himself isn't considered a prophet, but he is considered to be divinely inspired to collect and summarize what Christianity professes about the afterlife. The Comedy's version of the afterlife is also made the official standpoint of the Church about what comes after death.
Here is a possible POD:
Marco Polo returns from Asia with moveable type, something he observed used for printing chinese money. He sells these to an Italian entrepreneur who so happens to be a huge Dante fan. The man makes moveable type using steel instead of wood, and prints the first ever Bible (including 73 books), with all of Dante's works on the end.
This, of course, becomes a very lucrative business in Italy. To print the Bible, one needs a license from the Church, which is charged by the page. It became obvious that the Pope was more likely to give licenses for Bibles which just so happened to have illustrations and appendices (2 Esdras, 3 Maccabees, Prayer of Manasseh, and the Divine Comedy). So, no Bible was printed without the Divine Comedy.
Then, in the Council of Florence in the mid 15th century, it is decided that 2 Esdras, 3 Maccabees, Prayer of Manasseh, and the Divine Comedy, along with the Deuterocanon, are all Canon and must be in every printed Bible. The doctrinal statement is hammered out that the "Deuterocanon" and "Divine Appendix" are inspired by God inasmuch they present and accurately sum up divine doctrines, but they do not meet the criteria of being word-for-word the divine word of God, as the New Testament and Hebrew Old Testaments were. This, ironically, might have been the position of men like Augustine (who's Canon is somewhat confused and may rely on a rationale such as this.)
Now let's answer your questions:
How do you think would this influence the development of European culture and religion in the coming centuries?
The main differences are that Papal Infallibility might not become approved doctrine in the 19th century, and this might open the door for reunification with the Eastern Church in the 21st century. Papal Infallibility is simply a bridge too far for the East.
Also, the reformation would likely kick out Dante and have a natural disdain for him. He won't be taught as part of the humanities, simply because of aversion to his doctrines. Just try arguing with a Protestant that Wisdom of Solomon is Scripture, even though it clearly defies the natural order of things by containing accurate prophecy, and they will not even address the issue because in a sense Protestants have infallibly declared that the Masoretic Text of the Old Testament is Canon.
Lastly, Catholics would probably end up allegorizing Dante, arguing that he was a poet and he defies literal interpretation--his cosmology being off in the Paradiso would be an obvious reason to take this stance. Such a view negates claiming he was inspired to begin with, of course, but Dante's view of limbo, and a literal prolonged stay in purgatory might not match with what moderns are comfortable with believing. His view of heaven, however, may stick. Instead of the "angels playing harps on clouds" we might literally think of the beatific vision, and God being represented as being concentric circles, and etcetera.
Would it butterfly away the Reformation and maybe the Industrial Revolution and make more aspects of pre-Renaissance medieval culture survive longer?
No, it would hasten the Reformation in the north because the printing press is invented sooner. The Reformation would look more like a revolution against the Pope and a rejection of the Council of Florence, so Protestantism might look more like Eastern Orthodoxy. Baptists and Presbyterians would never exist within the pale of Protestantism, and would likely be considered more like Mormons and Adventists today.
The Roman Church by default will become 10 times more anti-science as Copernican astronomy would call into question Dante's divine inspiration, as well as other Biblical texts. So, expect there to be a kind of dark ages in Catholic Europe (which may shrink to just Italy, Spain, and part of the Balkans) and France, Great Britain, and Germany would be Protestant.
So, pre-renaissance backwardness, with a love affair for Platonism and anything backward, would continue within Catholic Europe. The rest of Europe would move along as is, perhaps without the 30 years war and what not as Protestantism would be more ecclesiologically similar and less extreme, maintaining sacraments and thus not causing the sort of rift which makes people really think they are going to hell over a doctrine.
How would the present day look like in this TL?
Protestantism I think would develop somewhat, but would retain elements of sacralism (state churches, enforced punishments of heterodoxy). For example, it homosexual marriage may be legal today in the west, but the building of Presbyterian Churches would not be until 75 years ago perhaps.
Italian, Portuguese, and Spanish culture will eventually accept that Dante's cosmology is not literal, but the years of backwardness would make their cultures even more quaint. Ironically, people may blame Dante and not economics for their backwardness, which in some ways would be an untruth because in the end of the day astrology did not make or break spain and italy.
Roman Catholicism would still be the world's largest denomination, because of Spanish and Portuguese colonizing and conversions. Eastern Orthodoxy might just be considered "Orthodoxy" and Anglicans, French Catholicism, German Catholics, and etcetera would be considered Apostolic CHurches in their own rights. All would quibble over doctrines but consider themselves historical churches of God. I imagine diversity within the doctrines of these churches, so some of these would allow for female priests, and people not going to hell for not going to confession--you know, the worse parts of liberal Christianity.