Ttyjukl

1826: The Last Man, a novel by Mary Shelley, available at Project Gutenberg.

From the Wikipedia page:

The Last Man is a post-apocalyptic science fiction novel by Mary Shelley, which was first published in 1826. The book tells of a future world that has been ravaged by a plague.

The story is set entirely in the future (late 21st century) unless you count the introduction where she pretends to explain how she found it. Quoting Wikipedia again:

Mary Shelley states in the introduction that in 1818 she discovered, in the Sibyl's cave near Naples, a collection of prophetic writings painted on leaves by the Cumaean Sibyl. She has edited these writings into the current narrative, the first-person narrative of a man living at the end of the 21st century, commencing in 2073 and concluding in 2100. Despite the futuristic setting, the world of The Last Man appears to be relatively similar to the era in which it was written.

Here is another, much later, story; not in the running for "first", but maybe you will like it better because it features a much more advanced future, with interplanetary travel, etc.

1915: "John Jones's Dollar", a short story by Harry Stephen Keeler, first published in the August 1915 issue of The Black Cat, available at Project Gutenberg. I quote the beginning of the story:

On the 201st day of the year 3221 A.D., the professor of history at the University of Terra seated himself in front of the Visaphone and prepared to deliver the daily lecture to his class, the members of which resided in different portions of the earth.

The instrument before which he seated himself was very like a great window sash, on account of the fact that there were three or four hundred frosted glass squares visible. In a space at the center, not occupied by any of these glass squares, was a dark oblong area and a ledge holding a piece of chalk. And above the area was a huge brass cylinder; toward this brass cylinder the professor would soon direct his subsequent remarks.

In order to assure himself that it was time to press the button which would notify the members of the class in history to approach their local Visaphones, the professor withdrew from his vest pocket a small contrivance which he held to his ear. Upon moving a tiny switch attached to the instrument, a metallic voice, seeming to come from somewhere in space, repeated mechanically: "Fifteen o'clock and one minute—fifteen o'clock and one minute—fifteen o'clock and one min—" Quickly, the professor replaced the instrument in his vest pocket and pressed a button at the side of the Visaphone.

As though in answer to the summons, the frosted squares began, one by one, to show the faces and shoulders of a peculiar type of young men; young men with great bulging foreheads, bald, toothless, and wearing immense horn spectacles. One square, however, still remained empty. On noticing this, a look of irritation passed over the professor's countenance.

But, seeing that every other glass square but this one was filled up, he commenced to talk.

"I am pleased, gentlemen, to see you all posted at your local Visaphones this afternoon. I have prepared my lecture today upon a subject which is, perhaps, of more economic interest than historical. Unlike the previous lectures, my talk will not confine itself to the happenings of a few years, but will gradually embrace the course of ten centuries, the ten centuries, in fact, which terminated three hundred years before the present date. My lecture will be an exposition of the effects of the John Jones Dollar, originally deposited in the dawn of civilization, or to be more precise, in the year of 1921—just thirteen hundred years ago. This John Jon—"

At this point in the professor's lecture, the frosted glass square which hitherto had shown no image, now filled up. Sternly he gazed at the head and shoulders that had just appeared.

"B262H72476Male, you are late to class again. What excuse have you to offer today?"

From the hollow cylinder emanated a shrill voice, while the lips of the picture on the glass square moved in unison with the words:

"Professor, you will perceive by consulting your class book, that I have recently taken up my residence near the North Pole. For some reason, wireless communication between the Central Energy Station and all points north of 89 degrees was cut off a while ago, on account of which fact I could not appear in the Visaphone. Hence—"

"Enough, sir," roared the professor. "Always ready with an excuse, B262H72476Male. I shall immediately investigate your tale."

From his coat pocket, the professor withdrew an instrument which, although supplied with an earpiece and a mouthpiece, had no wires whatever attached. Raising it to his lips, he spoke:

"Hello. Central Energy Station, please." A pause ensued. "Central Energy Station? This is the professor of history at the University of Terra, speaking. One of my students informs me that the North Pole region was out of communication with the Visaphone System this morning. Is that statement true? I would—"

A voice, apparently from nowhere, spoke into the professor's ear. "Quite true, Professor. A train of our ether waves accidently fell into parallelism with a train of waves from the Venus Substation. By the most peculiar mischance, the two trains happened to be displaced, with reference to each other, one half of a wave length, with the unfortunate result that the negative points of one coincided with the positive points of maximum amplitude of the other. Hence the two wave trains nullified each other and communication ceased for one hundred and eighty-five seconds—until the earth had revolved far enough to throw them out of parallelism."

"Ah! Thank you," replied the professor. He dropped his instrument into his coat pocket and gazed in the direction of the glass square whose image had so aroused his ire. "I apologize, B262H72476Male, for my suspicions as to your veracity—but I had in mind several former experiences." He shook a warning forefinger. "I will now resume my talk."

"A moment ago, gentlemen, I mentioned the John Jones Dollar. Some of you who have just enrolled with the class will undoubtedly say to yourselves: 'What is a John Jones? What is a Dollar?'

Micromégas (Voltaire, 1752)

This comes from the "starts on another planet" angle in your question rather than the "starts in the future" angle. I'm not sure if it fits that exactly anyway, since they come to Earth about halfway through the story. The protagonist, Micromégas, is a native of a planet orbiting Sirius, and the narrative begins with Micromégas on his home planet:

On one of the planets that orbits the star named Sirius there lived a spirited young man, who I had the honor of meeting on the last voyage he made to our little ant hill. He was called Micromegas[1], a fitting name for anyone so great. He was eight leagues tall, or 24,000 geometric paces of five feet each.

According to the plot summary on Wikipedia, Micromégas is found guilty of heresy and banished from court in his country. Out of frustration, he begins to travel the galaxy, and meets a resident of Saturn. They compare their experiences on their planets and ultimately travel together to Earth, where they explore for a bit and talk to some philosophers.

1863 Jules Verne wrote Paris au XXe siècle (Paris in the Twentieth Century) in 1863, but it wasn't published until 1994, so I'm not sure it counts.



Unlike Shelley's The Last Man, which has a reference to the writer's present in the introduction, this novels seems to be set entirely in the future of 1960.



From Wikipedia:

The novel's main character is 16-year-old Michel Dufrénoy, who graduates with a major in literature and the classics, but finds they have been forgotten in a futuristic world where only business and technology are valued. Michel, whose father was a musician, is a poet born too late.

The Battle of Dorking by George Tomkyns Chesney, published in 1871, starts several decades in the future and then flashes back to a time a few years after its publication date. It is thus all set in the future, although most of it is not very far in the future.

To answer my own question; the 1909 story The Machine Stops by E. M. Forster, which starts in an utopian or dystopian future (some seems to love their society, others don't).

No claim that it is the earliest, but Rudyard Kipling's "With the Night Mail (A STORY OF A.D. 2000)" is copyright 1905.

"In the Year 2889" by Jules and Michel Verne was published in 1889. It is about a day in the life of Fritz Napoleon Smith, a newspaper magnate, and all the technologies he uses. You can get a sense of its tone from the opening paragraph:

Little though they seem to think of it, the people of this twenty-ninth
century live continually in fairyland. Surfeited as they are with
marvels, they are indifferent in presence of each new marvel. To them
all seems natural. Could they but duly appreciate the refinements of
civilization in our day; could they but compare the present with the
past, and so better comprehend the advance we have made! How much fairer
they would find our modern towns, with populations amounting sometimes
to 10,000,000 souls; their streets 300 feet wide, their houses 1000 feet
in height; with a temperature the same in all seasons; with their lines
of aërial locomotion crossing the sky in every direction! If they would
but picture to themselves the state of things that once existed, when
through muddy streets rumbling boxes on wheels, drawn by horses—yes, by
horses!—were the only means of conveyance. Think of the railroads of
the olden time, and you will be able to appreciate the pneumatic tubes
through which to-day one travels at the rate of 1000 miles an hour.
Would not our contemporaries prize the telephone and the telephote more
highly if they had not forgotten the telegraph?

From page 75-76 of The History of Science Fiction by Adam Roberts, viewable on google books here, two possible very early examples:

Francis Cheynell's six-page political squib Aulicus: His Dream of the King's Sudden Coming to London (1644) is sometimes described as the first published future fiction. This oversells it, as there are plenty of prior examples of this mode of writing. More, Cheynell's text is something of a squib. Aucilus—Latin for courtier—is a dunce, unable to comprehend what has happened to his country, or the failings of the king he slavishly serves. Cheynell retells his dream of Stuart victory only to mock its absurdity. A companion volume Aulicus, His Hue and Cry Sent Forth After Britanicus, who is Generally Reported to be a Lost Man (1645) reflects back upon the Civil War, and finds rumours of Britain's 'loss' much exaggerated. Indeed, Aulicus is a stock figure in 17th-century satire, a means of critiquing court politics by invoking a foolish courtier without treasonously, and perilously, satirising the monarch himself.

Jacques Guttin's Epigone, histoire du siècle futur (1659) is a more considered attempt to portray a future society, although the flavour of the whole is very far from what we might consider futuristic. On the contrary indeed, Guttin's book rehearses romance and epic tropes in a fully nostalgic manner. The frame narrative describes how Epigone (whose name means posterity or after-born) and his friends are caught in an enormous storm at sea, are shipwrecked on the apparently African coast of Agnotie (Guttin's own glossary defines this as terre inconnue). They are taken inland to a mighty city, where by virtue of a crystal translation artefact they are able to communicate with the strange natives, to whose monarch Epigone relates his various adventures. Despite being notionally located in the future age, this text draws heavily on Virgil's Aeneid. The adventures themselves, including an interlude in a female-ruled amazonian kingdom devoted to sensual pleasure, echo Homer's Odyssey. The reversals, adventures, love elements, sword fights, escapes and all the usual bag and baggage of conventional romance dilute the notional futurity. Paul Alkon argues that 'Guttin's "future century" is not unequivocally a future at all' [Alkon, Origins, 37].