The Cask of Amontillado takes place in Italy during Carnevale: a festive time in the country similar to Mardi Gras in the United States. We start there, at night, in the madness, but are then taken back to the home of Montressor, more specifically, into the catacombs/wine cellars below. The setting is described as dark and damp, with niter climbing the walls and a mix of casks of wine and bones littering the area. The men carry flambeaux, creating the idea of darkness with only the small light of fire guiding the way. The Carnevale setting provides irony of a horror story taking place in such a festive and unlikely backdrop. 

Montresor lives in a palazzo. Fortunato lives in a palazzo. There must be plenty of such enormous buildings, and therefore the locale in question would have to be a large, important city. It would have to be in Italy because palazzo is an Italian word. It would have to be a seaport to explain the arrival of a shipment of Amontillado. Amontillado is only produced in Spain. The cask is described by both Fortunato and Montresor as a "pipe." A pipe contains 126 gallons. This is an enormous barrel. A shipment of such barrels could not have been transported over the mountains between Spain and France and then across France and over the Alps into Italy in wagons drawn by mules. It must have arrived by ship from Barcelona. Venice is the only possible destination, since Venice is full of old palazzi built in its former days of glory, and since Venice is still famous for its annual carnival. Neither Montresor nor Fortunato intends to drink all that Amontillado. They see it as an investment. They can store it indefinitely in oak barrels and it will only improve with age. They can bottle it and sell it off in cases to the British and Austrian millionaires Montresor mentions in the third paragraph of the story.

Henry James writes about an old palazzo in his story "The Aspern Papers."  

I forget what answer I made to this--I was given up to two other reflections. The first of these was that if the old lady lived in such a big, imposing house she could not be in any sort of misery and therefore would not be tempted by a chance to let a couple of rooms. I expressed this idea to Mrs. Prest, who gave me a very logical reply. "If she didn't live in a big house how could it be a question of her having rooms to spare? If she were not amply lodged herself you would lack ground to approach her. Besides, a big house here, and especially in this quartier perdu, proves nothing at all: it is perfectly compatible with a state of penury. Dilapidated old palazzi, if you will go out of the way for them, are to be had for five shillings a year. And as for the people who live in them--no, until you have explored Venice socially as much as I have you can form no idea of their domestic desolation. They live on nothing, for they have nothing to live on."                                                                                        Henry James, “The Aspern Papers” (1888)

Montresor is a poor man and apparently no longer has a family. He is probably only living in a palazzo with a skeleton staff of servants because the rent is cheap and the big building helps him put on a good "front."

That it is an ancient city is shown by the fact that so many centuries of accumulated human bones are described as the two men wind their way through the catacombs. Montresor did not attach the chains to the granite wall. They had been there for centuries and had been used for the same purpose by feudal lords to punish rebellious subjects.

Edgar Allan Poe's classic short story is set some time before 1846 (the year it was first published) and quite possibly during the late 18th century. The story is almost certainly set in Italy, although no specific city is mentioned; but it could also take place along the Italian-French border, since Mardi Gras-type carnivals were more common in France at the time. It was not unusual for Poe--a distinctly American writer--to use a European setting.

... Poe believed in using elements from Europe if they were useful artistically, and he believed that international settings helped establish universality... In ‘‘The Cask of Amontillado,’’ therefore, he used a European setting to create his exotic and murky atmosphere... (eNotes, "TCOA," Historical Context) 

We know it takes place during "the supreme madness of the carnival season," and most of the story evolves beneath the home of Montresor in the family catacombs that doubles as a wine cellar. Poe sets his entire tale at night, which adds to its creepy uncertainty and the impending evil that unfolds. Poe seems to deliberately keep the locale unknown to the story's readers, muddling it with Italian words and characters (Fortunato, Luchesi), Latin phrases, a French killer (Montresor), and a Spanish wine.  

If Poe's readers could not be expected to identify the nationality of each element, so much the better for creating the impression that the story happens "in another place and time.’’  (eNotes, "TCOA," Style) 

Poe leaves the setting ambiguous, but internal clues point toward a Southern location, whether in Europe or America. The names "Montresor" and "Fortunato" are from one of the Latin languages (Spanish, French, or Italian), and so could indicate the city is set in one of those countries. American writers of the time had a fascination with southern Europe (as did the British), and it was frequently used as settings in Romantic and Gothic works.

The fact that there was a carnival also indicates Mardi Gras, usually celebrated in predominantly Catholic locations. New Orleans in the US is an obvious choice for that one, but other Mediterranean locales would also answer.

As with much in this story, Poe leaves much unanswered.

The story takes place during Carnival/Mardi Gras celebrations (possibly in New Orleans?).  Montresor meets Fortunato as the latter is celebrating, perhaps a little too much.  The alcohol mixed with party atmostphere make Fortunato more easily led in search of this fabulous wine. 

From the life of a party, down to the depths of the earth in the crypts below Montresor's home, the setting symbolizes Fortunato's journey from life without cares to his death at the hands of Montresor.  The creepiness of the the crypts, with cobwebs, loose mortar, etc. juxtapose the gaity of the carnival, providing an even greater contrast lending itself to the eerie mood.

The specific setting of the story is never actually stated.  The events would appear to have taken place in a European country, but while "Fortunato" and "Luchesi" are Italian names, "Montressor" is most arguably French, and "Amontillado" is a Spanish wine.  Also, Montressor's coat of arms is Scottish in origin, and there are various Latin phrases and references scattered throughout the narrative.  The exact location of the story is ambiguous, leaving the impression only that it is "another place and time".

The exact timing of the story is never stated either, although critics have most often placed it in the eighteenth or nineteenth century.

In a more general sense, the story is set in a city, again, most likely European, and a long time ago, during a season of carnival or celebration.  The majority of the action takes place in the labyrinth-like, dreary catacombs, or series of underground chambers, underneath the palazzo, or estate, of Montressor.