What beliefs does the author ask you to suspend in order to appreciate and understand his story?

In order to appreciate this short story, you would have to suspend several disbeliefs. This, of course, depends on your beliefs prior to reading the story, including religious ones.
The first and most obvious of these would be the existence of the devil himself, presented to the reader as Mr. Scratch. As a reader, you must suspend your disbelief in the devil (if this applies to your life) in order to benefit from the story.
The next would be the existence of human souls, since Daniel Webster’s client sold his soul in exchange for good luck.
Then, one would need to suspend disbelief about the ability of ghosts to appear before living people, as the jury of the damned does during the trial.
Finally, one would need to suspend disbelief in the ability to predict the future, since Scratch accurately foretells Webster’s political trajectory and life experiences.