When Moshiach arrives and the Beis Hamikdash is Rebuilt and the Avodah starts again will we be required to fast on Yom Kippur? (please provide sources for your answers)
This does not mean that the festivals in the Torah will be abolished, but that the appreciation of the miracles and the redemption that occurs when the Mashiach comes and the temple is rebuilt will overshadow them. However, the Halachos of the Torah (including Yom Kippur) will continue.
As an example chabad says
The Midrash (Mishlei 9:2) teaches: "All the festivals will be annulled in future time, except for Purim." The future revelation of Divinity will be so intense that the revelation currently evinced by the festivals will be as insignificant as a midday candle. Purim, however, will be the exception, because the Purim miracle was called forth by the year-long self-sacrifice of the Jewish people of that time. (They could have averted Haman's decree by apostasy.) Their self-sacrifice evoked a Divine reaction so sublime that even in the future time it will never be annulled.
The gemoro in Rosh Hashana 18b discusses what will happen to the fasts decreed for the destruction of the temple after the temple is rebuilt and Bnei Yisroel are free of the oppression of the nations. These are the only fasts that will be changed to feasts or will be annulled.