No, there seems to be no evidence that Poe was taphephobic.
This answer relies heavily on the master's thesis Taphephobia in Edgar Allan Poe's Collection of Gothic Tales: A New Historicist Study of 19th Century America's Most Prevalent Fear, written by Salma Layouni as part of her MA degree at the University of Sousse, Tunisia. If you really want to know more about taphephobia in Poe's work, this long and detailed study is worth reading.
Taphephobia was a common thing among the general public
in Poe's time. This Wikipedia page
describes the history of the fear and how it led to the invention of "safety coffins" in which one could survive being buried alive:
The fear of being buried alive peaked during the cholera epidemics of the 18th and 19th centuries but accounts of live burial have been recorded even further back. [...] The general fear of premature burial led to the invention of many safety devices which could be incorporated into coffins. [...] A large number of designs for safety coffins were patented during the 18th and 19th centuries and variations on the idea are still available today.
So it makes sense that he would choose this as one of the many topics and themes he wrote about, regardless of his own feelings. Good horror stories play on people's existing fears, and if taphephobia was a prevalent fear among his readership, then it's a good choice to write horror stories about. He even discusses, in the opening paragraphs of "The Premature Burial", why he chose this topic, and plays upon the fears of his readers while doing so:
To be buried while alive is, beyond question, the most terrific of these extremes which has ever fallen to the lot of mere mortality. That it has frequently, very frequently, so fallen will scarcely be denied by those who think. [...]
It may be asserted, without hesitation, that no event is so terribly well adapted to inspire the supremeness of bodily and of mental distress, as is burial before death. The unendurable oppression of the lungs--the stifling fumes from the damp earth--the clinging to the death garments--the rigid embrace of the narrow house- the blackness of the absolute Night--the silence like a sea that overwhelms--the unseen but palpable presence of the Conqueror Worm--these things, with the thoughts of the air and grass above, with memory of dear friends who would fly to save us if but informed of our fate, and with consciousness that of this fate they can never be informed--that our hopeless portion is that of the really dead--these considerations, I say, carry into the heart, which still palpitates, a degree of appalling and intolerable horror from which the most daring imagination must recoil. We know of nothing so agonizing upon Earth--we can dream of nothing half so hideous in the realms of the nethermost Hell.
The fact that he identifies himself with the taphephobe, in his consistent use of "we" in this passage, serves the dual purpose of making the readers feel closer to him and of setting the scene for his first-person narration of a premature burial - in no way does it suggest that he himself was taphephobic.
Layouni discusses at length (too much length to summarise here) how "The Premature Burial" is designed in such a way as to play on the fears of a typical American:
The realistic feature of this particular tale grants its historical dimension that can be used to study a social phenomenon that invades the 19th C American society. Poe uses different literary techniques to guarantee different effects on different readers and to widen the scope of taphephobia as a universal fear that can overwhelm any human being.
If this is true, it seems less likely that the story was based on the author's personal fear, rather than just a prevailing fear in the society of the time.
In addition, there were already various existing stories about being buried alive
on which, it seems, at least in part, Poe based some of his own such stories:
- "The Premature Burial" is very similar in plot to John Galt's Blackwood story "The Buried Alive" (1821) - a comparison of the two may be found in Andrew Mangham's article "Buried Alive: The Gothic Awakening of Taphephobia" from the Journal of Literature and Science.
- Parts of "The Premature Burial" are also very similar to the notes and poems accompanying a "life-preserving coffin" exhibited at the annual fair of the American Institute in 1843 - the many verbal parallels are discussed in W.T. Bandy's A Source of Poe's "The Premature Burial" and T.O. Mabbott's annotated The Collected Works of Edgar Allan Poe (1978).
- "The Cask of Amontillado" is very similar to Joel T. Headley's "A Man Built in the Wall", published in the Columbian Magazine in August 1844 - I haven't been able to find this story online, but an excerpt can be seen in T.O. Mabbott's annotated The Collected Works of Edgar Allan Poe (1978).
- "The Cask of Amontillado" could also be partly inspired by a nun's interment in the poem Marmion by Walter Scott - a writer whose work is known to have inspired Poe on other occasions.
- "The Cask of Amontillado" could also be partly inspired by the 1831 short story "La Grande Bretèche" (which I've also read, and they are indeed reminiscent of each other) by Honoré de Balzac - another writer who is often compared to Poe and may have inspired him more widely.
It's clear, then, that taphephobia was already a theme in the literature of the time. Poe was following an existing trend, both in real people's beliefs and in horror stories, and there's no particular reason to think he was inspired by his own personal fears about being buried alive.
Interestingly, while writing this answer, I came across a new tidbit of information which is almost
what you're looking for: that Poe was afraid of the dark
. The following quote is from Home Life of Poe
by Susan Archer Weiss, who knew him in his last few years:
Mr. John Mackenzie, in speaking of Edgar, bore witness to his high spirit and pluckiness in occasional schoolboy encounters, and also to his timidity in regard to being alone at night, and his belief in and fear of the supernatural. He had heard Poe say, when grown, that the most horrible thing he could imagine as a boy was to feel an ice-cold hand laid upon his face in a pitch-dark room when alone at night; or to awaken in semi-darkness and see an evil face gazing close into his own; and that these fancies had so haunted him that he would often keep his head under the bed-covering until nearly suffocated.
So he did have a phobia, which he might have channelled into writing some of his "buried alive" stories (like the protagonist of "The Premature Burial", he would have found it unpleasant to wake up in a dark and unfamiliar place), but it wasn't specifically taphephobia.
Your question is, perhaps, a tribute to Poe's brilliance. His stories are so well-written that you wonder if they're based on his own fears, if they have that almost inimitable sign of a deep personal connection from the writer, rather than just being a great portrayal of phobia which evokes that fear in the reader.