Dante and the church had a rather antagonistic history even well before his writing of the Commedia, which took place from roughly 1315 until his death in 1321. In 1301, Pope Boniface VIII orchestrated an intervention in Florentine politics that led to the downfall of the White Guelphs, a party of which Dante was a leading member and which had been advocating for decreased papal influence. This led to Dante's permanent exile from Florence and inflamed his enmity against Boniface (and papal authority) even further. In 1302, the Pope issued the bull Unam sanctam, which essentially argued that all people, but most importantly secular Catholic rulers, were ultimately subject to papal authority. This was predictably explosive: a year later, Boniface died in the captivity of King Philip IV of France.

Dante's political treatise De Monarchia, probably written around 1318 (though this dating is not without controversy), is notable for its strident criticism of Unam sanctam. It was this treatise, rather than his poetry, that attracted the most significant amount of immediate opprobrium from the church, but the Commedia was (to a lesser degree) guilty by association. There are also important elements in the poem that are indicative of Dante's issues with the church and its hierarchy; most obvious is the fact that he represents a number of Popes (including Boniface) as condemned to Hell in the Inferno. Joan M. Ferrante, in her article "Hell as the Mirror Image of Paradise," writes:

That the church also knew the poem and understood its critical message is clear from various attacks: a Dominican, Guido Vernani, called Dante's poetry "a poisonous vessel of the father of lies, covered with false and fallacious beauty, by which the author, with poetic phantasms and figments and the eloquence of his words, his siren songs, fraudulently leads not only the sick and ignorant but even the learned to destroy the truth which might save them" (De reprobatione Monarchiae). The reading of Dante's poetry was prohibited at a Dominican chapter in Florence in 1335, although it apparently continued to be popular among the brothers; and various passages from the poem—on ecclesiastical corruption—were condemned by church inquisitions. The whole of the Monarchia was on the papal index [of forbidden books] from 1554 to 1881.

The Commedia was less vilified in ensuing decades than the Monarchia, though the ecclesiastical reputation of Dante and his works was definitely skating on some thin ice. Vernani doesn't necessarily speak for the whole church, and his tract (as its title indicates) is primarily focused on attacking De Monarchia and re-emphasizing the primacy of the church. Teodolinda Barolini notes in her book Dante and the Origins of Italian Literary Culture that "the church on the whole (with a few telling exceptions like the Dominican ban of 1335) was willing to bracket Dante as a poet, a maker of fictio." Anthony K. Cassell, in his book The Monarchia Controversy: An Historical Study, writes:

Although no official ever made a posthumous indictment of heresy against Dante, the perpetual dread that his departed father could be liable to such an accusation haunted Dante's son Pietro throughout his long and prosperous life. The possibility that his father's Commedia could be found heterodox and that he himself could forfeit his patrimony and wealth drove Pietry to compose three versions of his commentary on the sacro poema to explain away its controversial verses.

In later centuries, as the Monarchia receded into obscurity and the Commedia enjoyed a resurgence of popularity, the latter would sometimes be condemned by the church primarily due to its depiction of several popes residing in Hell, which (as I understand it) runs counter to Catholic doctrine.

TL;DR: In the short term, the church was somewhat offended, but not tremendously or uniformly so. Dante's treatise De Monarchia was seen as more dangerous due to its opposition to extreme papal authority, but even that did not earn him the label of heretic.