What point of view is used in Poe's "The Pit and the Pendulum"?

This story is told from the first person limited point of view.  The narrator is a prisoner being held in captivity during the Spanish Inquisition until conquering soldiers come and rescue him.

The fact that the narrator doesn't know much about his surroundings while in the pit and has to grope around to "visualize" his situation increases the sense of horror (especially when he almost falls into the pit unaware).

Check out the following references for more details concerning Poe's use of point of view in this particular short story to create effect.

"The Pit and the Pendulum" by Edgar Allan Poe is told in the first-person limited point of view, and we know it from the first sentence in the story:

I was sick, sick unto death, with that long agony, and when they at length unbound me, and I was permitted to sit, I felt that my senses were leaving me. 

The narrator is telling the story, as it happens, from his point of view, thus the use of the pronouns "I," "me," and "my." 

The benefit of Poe's using this point of view for this particular story, of course, is that we get to experience the same fears and horrors as he does. If anyone else would have told this story, we would have gotten the facts, but we would lose the emotions and thinking which the narrator shares with us. Even an omniscient narrator would not have given us the same effect as a narrator who is telling his own story. Of course, the events of the story, connected to the Spanish Inquisition, are horrific; they are made even more so as we experience them through the eyes, ears, and mind of the victim.

How does Edgar Allan Poe develop the first person point of view and narrative in "The Pit and the Pendulum"?

Edgar Allan Poe's superb short story "The Pit and the Pendulum" has long been subject to many existential and Freudian interpretations.  In the vacillation of the first-person unreliable narrator who is hallucinatory at times and rational in others, there are shifts between the subconscious and the conscious mind.  Thus, the narrative mode is one of psychic reflections and few realities:

Amid frequent and thoughful endeavors to remember, amid earnest struggles to regather some token of the state of seeming nothingness into which my sould then lapsed, there have been brief, very brief periods when I conjured up remembrances which my lucid reason assures me could refer only to that condition of seeming unconsciousness.

The narrator's variable level of consciouness directs the narrative as his perception of things such as the candles and the dimensions of the cell differ depending upon his mental state. The "shadows of memory" recall motion and sound while his subconscious senses the "horror at my heart" and a "rushing revival of soul and a successful effort to move" as his conscious mind has

a full memory of the trial, of the judges, of the sable draperies, of the sentence, of the sickness of the swoon.

It is important to keep in mind that the narrator tells his experiences in retrospect.  And it is this retelling by the "sick unto death with that long agony" narrator that contributes to the horror of the tale.  The hallucinatory quality of the narration also contributes to the Gothic effect of Poe's tale of transcendent experience.

Narrative mode is the collection of choices an author makes about how to tell the narrative. Narrative mode includes point of view and voice (i.e., narrative point-of-view and narrative voice).

Narrative point-of-view ...

...determines through whose perspective the story is viewed [focused].

Narrative voice is "manner through which the story is communicated" to the reader by the author: casual, formal, etc.

First person point of view is defined as when...

...[the] narrator participates in action...

The narrative point of view is an aspect of the narrative mode, which the "author uses to convey the plot to the audience." The narrative mode also...

...encompasses not only who tells the story, but also how the story is described or expressed...

This is key to your question: in first-person, a suffering man tells the story of his torture, but the narrative mode provides more intellectual clarity—as the story moves between what seems to be insanity (imaginings) to rationalism.

First-person narrative is defined as...

...personal point of view of the first person, usually the author participant if the writer assumes the point of view of a character.

In Edgar Allan Poe's "The Pit and the Pendulum," the narrator is a "participant" in the story, speaking in first-person point of view.

First-person narration draws the reader’s attention to the narrator.

The first person point of view may not always be reliable (especially in this story where the narrator is being tortured), while the narrative point of view delivers other details in a more logical manner. For example, the candles that the narrator describes in first-person may be hallucinations because of the narrator's torture:

And then my vision fell upon the seven tall candles upon the table. At first they wore the aspect of charity, and seemed white and slender angels who would save me; but then, all at once, there came a most deadly nausea over my spirit, and I felt every fibre in my frame thrill as if I had touched the wire of a galvanic battery, while the angel forms became meaningless spectres, with heads of flame, and I saw that from them there would be no help.

However, in the segment below, the narrator is more in control of his emotions here—delivered with an concerted effort—to rationally comprehend his situation: collecting facts about his condition and his surroundings...Where is he? What chances for survival are there? He is not giving in to hysteria but calculating—separating the real threat from the imagined danger.

The intensity of the darkness seemed to oppress and stifle me. The atmosphere was intolerably close. I still lay quietly, and made effort to exercise my reason. I brought to mind the inquisitorial proceedings, and attempted from that point to deduce my real condition. The sentence had passed; and it appeared to me that a very long interval of time had since elapsed. Yet not for a moment did I suppose myself actually dead. Such a supposition, notwithstanding what we read in fiction, is altogether inconsistent with real existence;—but where and in what state was I?

Poe moves back and forth between the first-person and narrative points of view—which mimics the mental processes of one who is suffering (physically, mentally or emotionally—creating illusions) on one hand, while struggling for coherence on the other. The danger of losing one's sanity is displayed for the reader when using the first-person point of view. On the other hand, in the narrative point of view, the narrator considers (rationally) what is consistent with "real existence."

This tactic guides the reader between fear of suffering, terror and insanity; but realism is maintained through details grounded in fact. This form is also used in Poe's "The Fall of the House of Usher."