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?
At the Time of the King Salomon, People followed the Beth Din instructions. Then, the Beth din realized his error, the Beth Din needed to bring a bull as special Qorban Chatat. The bans are only Chayave Kretot, see the first mishna of massechet kretot (actually, they cooked and ate).
We found something in Gemara Moed Qatan and Horayot about a verse.
Melachim I, 8, 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 R' Johanan, said that that year Israel did not observe the Day of Atonement, whereat they were perturbed
At the time of King Salomon they cooked and ate on Yom Kippur and they offer Par Heelem Davar Dhel Tzibur (Eating and cooking in Yom Kipur is Chayav Qaret). This is clear from the Gemara in Horayot 3a from the words of Rav Asi commenting a verse in Melachim 1.8.65. About the Par Heelem Davar Shel Tzibur, there are several opinions(Horayot 3a) who need to bring it. According to one opinion in the name of Rabbi Meir, the Beth Din bring it. Following Rabbi Yehuda the congregation, following Rabbi Shimeon Ben Eleazar in the name of Rabbi Meir, both the congregation and the Beth Din bring a Qorban Tzibur.