Will we celebrate Yom Kippur in the future when Moshiach comes?
The celebration of Yom Kippur will continue even after the coming of Moshiach. We will also fast on this day just as was done prior to Moshiach’s coming. The purpose of this fast will be to atone for sins performed when the Yetzer Hara was still intact. Nevertheless, the other forms of oppressions, such as not wearing leather shoes, not bathing and the prohibition against marital relations, will be rescinded.
During the seven days of the inauguration of the Temple: If the seven days of inauguration of the third Temple will fall during Yom Kippur, then on that Yom Kippur we will not fast, just as was done during the seven days of inauguration of the first Temple in the times of Shlomo. This is due to the Simcha required to be shown during the inauguration.
 Yalkut Shemoni Mishleiy 944 “All the holidays will be nullified in the future except for Purim. Rebbe Eliezer says even Yom Kippur will never be nullified as the verse states “Chukas Olam/an everlasting statute”
 Radbaz 2/828; Peri Tzaddik [Rebbe Tzadok of Lublin] 5 Motzei Yom Kippur 2
The reason: As the Torah is eternal and thus the notion that a Holiday will be nullified in the literal sense is not acceptable. The other Holidays will be nullified in the sense that their pleasure will not be felt due to the constant pleasure of the redemption era. However on Yom Kippur since we will fast, the oppression will be felt even more that today. [Radbaz ibid]
 Peri Tzaddik ibid
 Peri Tzaddik ibid based on Tikkunei Zohar 21 which states “Purim is called like Yom Kippur because in the future one will have pleasure on Yom Kippur and the day of oppression will be turned to joy.” He interprets this to refer to the other forms of oppression, other than fasting. Regarding the difference between fasting and the other oppressions [which are all Biblically forbidden], see Admur 611/2 that fasting is always called and oppression while the other matters are only occasionally called an oppression, and is hence only Biblically forbidden due to a tradition of the Sages.
 See Moed Katan 9a; Toras Menachem 1992 Erev Yom Kippur