That’s a popular conception of the Middle Ages, but “the church” couldn’t and didn’t dictate every aspect of life. Oh it definitely thought that it could! But that’s not really how it worked. It is true, though, that almost anyone with a good education would have been educated by the church in some way.

The other problem is…which church? I guess you probably mean the Latin church in Western Europe based in Rome. In that case the church didn’t really have any particular opinion about the Iliad (or the Odyssey), because they just didn’t know about it. There was no text of the Iliad known in Latin Europe. It was in Greek, and full texts of the Greek Iliad were certainly known and read and studied in the Byzantine Empire, but no Greek versions were known or translated into Latin in Western Europe until the 15th century.

That doesn’t mean Latin-educated people didn’t know about the Trojan War at all though. They knew about from other versions and other literature that mentioned the same characters. They had the Ilias Latina, a short summary of the Iliad, only 1070 lines, which was attributed to an ancient Roman translator, Publius Baebius Italicus. Clearly Italicus was working with the original Greek - I’ll have to add another layer of translation here, but here is George Kennedy’s translation of Italicus:

Inform me, Goddess, of the anger of Peleus' haughty son

That brought grim deaths upon the wretched Greeks

And sent brave souls of heroes to the underworld,

Leaving their limbs be tom by the jaws of dogs and birds;

In open fields their bloodless bones unburied lay,

Consequent upon the will of heaven's king

Unrivaled, when discordent hearts provoked a quarrel: those of the

Sceptered son of Atreus' and Achilles' famed in war.

And here is Richmond Lattimore’s translation of Homer:

Sing, goddess, the anger of Peleus' son Achilleus

and its devastation, which put pains thousandfold upon the Achaians,

hurled in their multitudes to the house of Hades strong souls

of heroes, but gave their bodies to be the delicate feasting

of dogs, of all birds, and the will of Zeus was accomplished

since that time when first there stood in division of conflict

Atreus' son the lord of men and brilliant Achilleus.

Italicus’ name is in the text but everyone seemed to have believed this was really written by Homer. It was a very popular text and was used for basic Latin instruction even for very young students, so apparently the church didn’t object to it (or even if it did, no one cared).

Two other popular accounts of the Trojan War were known in Latin in medieval Europe, attributed to Dictys Cretensis and Dares Phrygius (Dares is actually a character in the Iliad). They supposedly wrote eyewitness accounts in Greek that were later translated into Latin - the translations were attributed to ancient Roman authors as well but they are more likely from the late classical/early medieval period. They were believed to be even more reliable than Homer, since they were apparently older, in prose not in poetry, and were much longer than the Ilias Latina, which was otherwise the only work of Homer known in Latin Europe.

Of course Western Europeans were also quite familiar with the Iliad and Odyssey and other Greek myths from the works of Ovid and Virgil, among other Classical Latin authors. They were still quite popular in medieval Europe and since they were already in Latin they didn’t have to be translated. The works of Italicus/Homer, Dares, Dictys, Ovid, and Virgil were then adapted into other medieval stories about the Trojan War. Just as there were epic cycles of stories about King Arthur or Charlemagne, known as “matters” (the Matter of Britain, the Matter of France), there was also a “Matter of Rome” based on ancient Roman and Greek myths, and a subset of that was the “Matter of Troy”.

One popular medieval epic poem about Troy was written in England by Joseph of Exeter in the 12th century. But although Latin was the usual language of education, it wasn’t the only literary language, and the Matter of Troy was also read in French and English. The Roman de Troie was written by Benoît de Sainte-Maure in the 12th century, based on Dares and Dictys. In the 15th century, the very first book printed in English was a translation of a French history of the Trojan War.

Latin Western Europe was of course not the only place Homer was read in the medieval world - in the Greek-speaking Byzantine Empire in Eastern Europe, Homer had never been forgotten, and the Greek texts of the Iliad and Odyssey were still well-known and frequently studied, read, and copied. The earliest full copy of the Iliad is actually from the 10th century, quite far removed from Ancient Greece, but it shows that Homer was still widely read in the Byzantine world. (That manuscript is currently in Venice but it was only brought there from Byzantium in the 15th century.)

Any educated person in the Byzantine Empire would have been very familiar with Homer. Just to take one example that I conveniently have at hand, Anna Komnene quotes from Homer dozens of times in her Alexiad, a history of the reign of her father Alexios I. Niketas Choniates, a historian in the early 13th century, likewise includes numerous quotes from Homer in his chronicle.

I can’t think of any specific objections from either the Latin or Greek churches, although I’m sure someone at some point must have been opposed to studying pre-Christian heroes; but for the most part, the Trojan War was assumed to be real historical events, just as real as events in the Bible, and theologians and historians attempted to fit the Trojan War and other events from Greek mythology into Biblical history. Sometimes Greek gods and heroes were considered to be allegories for Christianity - God had revealed truths to the ancient Greeks, even if they didn’t fully understand them yet, but Christians could now see how they were really Christian parables all along. So the church was never opposed to studying ancient myths, because they could still teach Christians valuable lessons.

Sources:

George A. Kennedy, The Latin Iliad: Introduction, Text, Translation, and Notes (1998)

The Iliad of Homer, trans. Richmond Lattimore (University of Chicago Press, 1951)

Tamara F. O’Callaghan, “Tempering scandal: Eleanor of Aquitaine and Benoît de Sainte-Maure’s Roman de Troie” in Eleanor of Aquitaine: Lord and Lady, ed. Bonnie Wheeler and John Carmi Parsons (Palgrave Macmillan, 2003)

Marilynn Desmond, “Homer in the Latin West in the Middle Ages”, and Maria Mavroudi, “Homer in Greece from the End of Antiquity: The Byzantine Reception of Homer and His Export to Other Cultures”, both in The Cambridge Guide to Homer, ed. Corinne Ondine Pache (Cambridge University Press, 2020)