I am an art historian. Before I begin I need to clarify some terms. Historians and lay people typically use the word Renaissance to describe the time between the 14th and 17 century. However art historians have a much more narrow definition, usually considering the Renaissance to take place roughly between the 15th and early 16th century, while using the term Baroque to describe the 17th century. I am more familiar with the Baroque period.

Late Medieval and Renaissance

While the Renaissance is typically characterized as being a rebirth of interest in the poetry, art, and philosophy of classical Rome and Greece, medieval artists still showed something of an affinity for classical figures. A frequent motif in sculpture was depictions of the "9 worthies": 3 worthy pagans (Hector, Alexander, and Julius Caesar), 3 worthy Jews (Joshua, David, and Judas Maccabeus), and 3 worthy Christians (Arthur, Charlemagne, and Godfrey of Bouillon.) Such a depiction can be found on the Schöner Brunnen in Nuremburg, sculpted in the late 1300s The conflation of Classical and Christian themes in art is at least a few centuries older than the Renaissance itself.

During the Renaissance itself, art depicting pagans and pagan subjects was not only permitted by the church, but highly sought after by church officials. Raphael's School of Athens was painted in the Pope's quarters. While not a depiction of pagan myth itself, the School of Athens shows that even the Pope himself had admiration for pagan scholars.

Baroque

Many artists could be discussed in terms of the acceptability of pagan subjects in art, but the best example is Gian Lorenzo Bernini (1598-1680). Bernini was a sculptor and architect who worked closely with Roman Catholic clergy. He made significant contributions to the design of St. Peter's Basilica. Two of his most famous works Apollo and Daphne and The Rape of Proserpina were commissioned by Cardinal Scipione Borghese. Pagan subjects were far from taboo, even at the highest level of the Roman Catholic Church.

However, the Baroque period is also when many sculptures started to come under fire. In response to the Protestant Reformation, the Council of Trent enacted new standards for art: limiting what was "acceptable" in terms of nudity and depictions of sexuality. When you see fig leaves covering the genitals of Renaissance and Baroque sculptures, they were usually glued on as a result of the Council of Trent. The Council of Trent primarily affected specifically religious topics: sculptures of David, the Virgin Mary, etc. I am not aware of what measurable effects it had on art depicting pagan subjects, but it is still worth mentioning since your question pertained to institutional responses.

Other Thoughts and Conclusion

Things were different in the Northern Renaissance. Artwork depicting pagan subjects was far less common, but I am not educated enough on the topic to tell you why. Surely, the Protestant Reformation played a large part.

Edit I did remember at least two works of art from the Northern Renaissance and Baroque periods. Both depict the Judgement of Paris from the Iliad. One is by Lucas Cranach, 1528 and the other by Peter Paul Rubens, 1635

I am unsure whether there was an official stance on art depicting pagan subjects. A religious historian would be better able to answer such a question. As an art historian I can confidently tell you that, given the popularity of pagan subjects among even clergymen, I would say there doesn't appear to have been an institutional disapproval by the Catholic church to such subjects in practice.

Sources/Further Reading

Baroque and Rococo Art and Architecture, 1st Edition by Robert Neuman provides good insight on some of these issues, particularly the Council of Trent.

Northern Renaissance Art: Painting, Sculpture, the Graphic Arts from 1350 to 1575, 2nd Edition by James Snyder is an excellent look at the Northern Renaissance, where such artwork was less common. Edit: Typo fix

Gardener's Art Through the Ages, 14th or 15th edition is a widely used art history textbook that could provide some answers as well.