How does James Joyce criticize Father Flynn in his story "The Sisters"?

Joyce's depiction of Father Flynn in "The Sisters" is marked by hints, innuendoes, and vague suggestions. As such, we never get a full picture of who this man really was or what made him tick. The only portrait we have of him comes to us through the eyes of others, such as the narrator and the old priest's sisters. This means, inevitably, that what little information we are given about Father Flynn is partial, incomplete, and potentially unreliable.
Nevertheless, the allusive quality of Joyce's descriptions of Father Flynn does suggest a certain critical stance on the part of the author, both toward Flynn as a priest and as a man. To some extent, Father Flynn's paralysis is symbolic of the general cultural malaise that Joyce believed had hung over Ireland for centuries. Joyce also believed that one of the main components of this malaise was the malign cultural influence exercised by the Catholic Church over Irish society. Paralysis is a common theme throughout Joyce's stories, and it is clear from "The Sisters" where much of the blame ultimately lies for the cultural and intellectual stasis at the heart of Irish life.
At a number of points in the story, Joyce vaguely suggests that the relationship between Father Flynn and the young boy was potentially inappropriate:

Old Cotter looked at me for a while. I felt that his little beady black eyes were examining me but I would not satisfy him by looking up from my plate. He returned to his pipe and finally spat rudely into the grate."I wouldn't like children of mine," he said, "to have too much to say to a man like that.""How do you mean, Mr. Cotter?" asked my aunt."What I mean is," said old Cotter, "it's bad for children. My idea is: let a young lad run about and play with young lads of his own age and not be . . . Am I right, Jack?""

The use of ellipses after the words "not be" is suggestive. According to Old Cotter, there is something unhealthy and rather troubling about a young lad spending so much time in the company of an adult.
Later on, we discover that Father Flynn confessed to the boy that he was a simoniac. This is someone who has committed the sin of simony—the buying and selling of church privileges. At the very least, this would suggest greed and moral corruption on the part of the old priest.
Nevertheless, as Joyce's portrayal of Father Flynn is suitably ambiguous, we should proceed with caution before ascribing any definitive character evaluations to the author, negative or otherwise. Most, if not all, references to the priest in the story can be interpreted in a number of different ways. Consider this passage, for example:

Sometimes he used to put me through the responses of the Mass which he had made me learn by heart; and, as I pattered, he used to smile pensively and nod his head, now and then pushing huge pinches of snuff up each nostril alternately. When he smiled he used to uncover his big discoloured teeth and let his tongue lie upon his lower lip—a habit which had made me feel uneasy in the beginning of our acquaintance before I knew him well.

There are two ways in which we can examine this excerpt. On one hand, we could say that this shows Father Flynn to be rather sadistic: he smiles devilishly at a naive young boy as he struggles to recite the responses of the Mass. On the other hand, Father Flynn's smile could be one of recognition: his mind could be returning to the time in his own childhood when he had to go through the exact same ordeal. In fact, we know through his sisters that Father Flynn was of a nervous disposition himself, so it is perfectly plausible that his smiling at the young boy was perfectly harmless, even if it induced a profound sense of unease at the time.
Father Flynn's nervousness was the start of his paralysis—or so his sisters believe. One day he broke the chalice, a sacred vessel designed to hold the body and blood of Christ. There was nothing in the chalice when Father Flynn broke it, but the sisters remained convinced that this was the beginning of the end. It is the same in life, as in death: Father Flynn, lying in his coffin, has only a very loose grasp of the sacred Eucharist chalice.
Again, Joyce forces us to choose between two different but related interpretations. We could see Father Flynn as a symbol of the Catholic Church in Ireland, casting a pall of social and cultural paralysis over all who come into contact with it. Alternatively, we could see him as one of the Church's victims, his life stunted and ruined by a thankless devotion. Irrespective of whichever explanation we think is best, there can be little doubt that the main thrust of Joyce's criticism is firmly directed toward the Church itself. The question that remains is whether Father Flynn is somehow a victim of the Church or whether he is actively complicit in the spread of its malign influence upon his native land.