In masechet horayot we learn that korbanot must be brought if the beit din makes an erroneous decision that is followed by the people. Do we know of such a mistake that was made by a beit din? If so, what was it?
People followed the bet din instructions. Then, the beit din realized his error, the bet din needs to bring a bull as special korban Chatat. The bans are only Chayave Kretot, see the first mishna of massechet kretot.
sabbahillel explained that you ask about "לא היה ולא עתיד להיות" as the Gemara says in Sanhedrin about Ben Sorer Umore, Ir Hanidachat and Beit Hamenuga. Up to day I do not find any similarity. Note that for the 3 cited above reasons are offered to argue this. But here what may be the the reason, mistake is impossible?
But maybe that we'll find something.
Melachim I, 15, 65
וַיַּעַשׂ שְׁלֹמֹה בָעֵת-הַהִיא אֶת-הֶחָג וְכָל-יִשְׂרָאֵל עִמּוֹ קָהָל גָּדוֹל מִלְּבוֹא חֲמָת עַד-נַחַל מִצְרַיִם, לִפְנֵי יְהוָה אֱלֹהֵינוּ, שִׁבְעַת יָמִים, וְשִׁבְעַת יָמִים--אַרְבָּעָה עָשָׂר, יוֹם
So Solomon held the feast at that time, and all Israel with him, a great congregation, from the entrance Hamath unto the Brook of Egypt, before the LORD our God, seven days and seven days, even fourteen days.
Massechet Moed Katan 9A
R' Parnak, reporting ' `Johanan, said that that year Israel did not observe the Day of Atonement, whereat they were perturbed
See here in name of the ROGOTSHOVER that at the time of Shlomo they cooked and eat on yom kipur and they offer Par heelem davar shel tsibur (Eating in Yom Kipur is Chayav Karet).