The fact that Father Hoyt is the first pilgrim to tell us his story tells us that Hyperion is probably going to have a lot to do with religion. And the fact that Father Hoyt is a sniveling, weak-willed dish towel tells us that Hyperion's views of religion might not be all that positive.
Religion aside, out of all the pilgrims who tell a story, Lenar Hoyt is the one we learn the least about. Mainly, it's because his story primarily focuses on that of another: his mentor, Father Paul Duré. The Consul does describe Father Hoyt a bit prior to Chapter One. He's "a priest of the old-style Christian sect known as Catholic" (1.20), and he's in his early 30s, but looks older than that, as though he's been sick a long time. The Consul notes "the physical echo of the boy in the man" (1.21). What could have aged him prematurely?
The problem seems to be not that his faith is false—but that it might actually be true. The resurrection of Jesus is a huge part of the Catholic faith, and Hoyt discovers that resurrection after crucifixion does exist, and that it's not all that it's cracked up to be.
Seeing Father Duré's painful death and rebirth changes Hoyt. He hates the Bikura, the strange tribe Duré was studying, for what he believes they did to Duré. He's no longer the mild-mannered holy man he was. When his traveling partner executes the Bikura in cold blood, Hoyt says, "I didn't... didn't argue with him. I laughed" (1.720).
Not only is he crazy vindictive, but Hoyt now holds the cruciform, the strange artifact that kept Duré in an endless cycle of death and resurrection. In addition, the Bikura blessed him with one of his own. This means that one day, Hoyt, too, may be resurrected. And so might Duré. They might both experience what Christ did, and Hoyt is so not looking forward to it. Death may be not proud, but resurrection just might be worse.
After Hoyt tells his tale, he spends the rest of the story in a haze of ultramorphine, only chiming in occasionally as if to remind us that he still exists. He does seem to go into shock at the beginning of Chapter 5 after discovering Het Masteen's blood-splattered room. But other than struggling to carry Masteen's luggage—"the Consul carried one end […] while Lenar Hoyt puffed and panted under the other end" like an out-of-shape bellhop (5.131), he doesn't contribute much else to the story. The man of shaken faith suffers privately.
Once you read the harrowing Priest's Tale, you know exactly why. Father Hoyt begins the tale as a loyal Catholic, "bound by obedience and schooled in discipline" (1.150). After that, well, whether he believes it or not, he's bound to it. Being of the cruciform gives him no choice. Catholicism doesn't look all that great when its teachings are literal and you have no say in the matter.