So far the earliest future seems to be in The Partisan Leader: a Tale of the Future By Edward William Sidney. Published in 1836, set in 1847 eleven years later. That is 73 years earlier than the next future date suggested, 1920 in The King in Yellow. That is a great find by User 14111.
I want to add two notes:
Encouragement to seek for even earlier future dates than 1847 in early works of proto science fiction.
Mention of fictional works that have earlier future dates than 1847, but might not count as science fiction. Your mileage may vary, as they say in TV Tropes.
Science fiction set before 1847
The Mummy!: Or a Tale of the Twenty-Second Century by Jane Loudon, was published in 1827, nine years before The Partisan Leader: a Tale of the Future, but does not displace it as the front runner since it is set later, in 2126.
The Last Man by Mary Shelley, was published in 1826, ten years before The Partisan Leader: a Tale of the Future, but does not displace it as the front runner since it is set at the end of the 21st century - the date of 2092 for some events is mentioned in the summary.
The Mummy! is set 300 years in the future, so a science fiction or proto science fiction work publish before 1547 and set 300 years in the future would have an earlier future date than The Partisan Leader: a Tale of the Future.
The Last Man is set 266 years in the future. So a proto science story written before 1581 and set 266 years in the future would have an earlier future than The Partisan Leader: a Tale of the Future.
And if someone in 1650 wrote a futuristic story set 150 years in the future in 1800, that would be an earlier fictional future. If someone in 1692 wrote a story set only 100 years in the future in 1792 that would be an earlier fictional future. And if someone in 1745 or 1790 wrote a story set only 50 years in the future, in 1795 or 1840, that would be an earlier future than The Partisan Leader: a Tale of the Future.
Thus it is possible that someone who researches early works of proto science fiction could find a story set in an earlier fictional future date than 1847.
Other predictive fiction set before 1847
The question "What was the first story to be set in the future?" has some great answers.
January First-of-May says:
The classic book series Gargantua and Pantagruel, written in the 16th century (first book c. 1532, last book c. 1564), occasionally gives mention of how much time had passed. There is about one specific date in the entire story - an early event is said to have happened in 1420.
As it happens, the author did not seem to take much care of the dates, and when one adds it all up, by the time we get to book four the story is taking place in the early 20th century.
To be fair, I would not have mentioned it at all if the entire thing did not look quite science-fiction-y already (especially in the last two books).
So book one was published about 1532, the fourth book was published in 1552. And the fictional dates seem to progress from 1420 to the 20th century (1901-2000). Thus at some point during the first four books there should have be a lot of events happening after 1552 and before 1847.
Thus Gargantua and Pantagruel is a work of fiction that includes fictional events in future periods of the 16th, 17th, 18th, 19th etc. centuries. But is it a work of SCIENCE fiction with earlier future dates than 1847, or is it a work of FANTASY fiction or of HUMOROUS fiction with earlier future dates than 1847?
And he mentions an earlier work, the play Medea by Seneca the Younger. Lines 375 to 379 have often been interpreted as predicting the Age of Discovery over 1,400 year later:
There will come an age in the far-off years
when Ocean shall unloose the bonds of things,
when the whole broad earth shall be revealed,
when Tethys shall disclose new worlds
and Thule not be the limit of the lands.
Thus January First-of-May may have found 2 future dates earlier than 1847 in his answer to another question.
It depends on how much like science fiction Gargantua and Pantagruel and Seneca's Medea seem to us, and especially to dot_Sp0T.