There isn't much mention made of the Christian religion, specifically, in this story. The Devil is treated more as a folk character and force of nature, possibly because Irving took inspiration from German folktales during this phase of his career, and those stories inherent much from pagan myths.
Some religious elements that are mentioned are the implication that Tom, like all humans, has a soul, and that the Devil is in charge of "damning" and "burning" sinners, as indicated by the symbolic trees in the swamp with the names of prominent men on them.
The most specific mention of religion that is made concerns Tom's supposed reformation in old age, where he begins attending church regularly and with great zeal, and he begins reading a Bible regularly. Some relevant quotes here include:
The quiet Christians who had been modestly and steadfastly traveling Zionward, were struck with self reproach at seeing themselves so suddenly outstripped in their career by this new-made convert.
This indicates that the regular churchgoers, rather than truly seeing Tom for the sham that he is, are somewhat ashamed that they don't appear nearly as devout as he is.
Another significant religious reference;
He even talked of the expediency of reviving the persecution of quakers and anabaptists.
This is a reference to the practices of the Puritan culture which preceded the current one, and dominated the majority of Massachusetts society for about 50 years in the 1600s. Quakers and Anabaptists were Christian sects that the Puritans outlawed, and believed to be heretical.