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, a scan of which is available at the Internet Archive. The text of the story is 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?'
