Will the Kohanim (priests) be anointed with oil during the inauguration of the Third Temple, like they were at the inauguration of the Mishkan (Tabernacle)?

Answer:

There are various opinions brought in the sources regarding whether the Kohanim will be anointed in Moshiach's time. In one place, our sages decisively negate the idea that they need anointment, and compare the Kohanim to the altar itself. Just as the altar was anointed only at the inauguration of the Mishkan and since then has not been anointed, the priests, too, were anointed only once in the desert and will not be anointed again. As is written in Midrash, "This is the inauguration of the altar" - It was anointed this time but will not be anointed again... "This is the anointing of Aaron and the anointing of his sons" - this time, and they will not be anointed again forever... not even in the World to Come.

However, from other sources in the words of our sages, Nachmanides proves that there is room to believe that the Kohanim will be anointed: "Since their anointment has already been nullified from Aaron and his sons at the time of their death, and they will be renewed (resurrected) people, they will need to be sanctified like the first time."

Rabbi Joseph Chaim of Baghdad (the Ben Ish Chai) dwells at length on this issue in is book "Sod Yesharim" and writes like Nachmanides: "Something done to the body itself before death is nullified after death. When the body will be resurrected, it will be like a new body..."

Rabbi Joseph Chaim emphasizes that the requirement to anoint the Kohanim anew applies also to Aaron, even though "his body was not changed in the grave at all; it is as if he is still alive and like a person who is asleep, for our sages have said (Talmud Baba Basra 17:1) regarding Aaron, that no worm or maggot had mastery over him. Only a moment before the resurrection of the dead will he return to dust. How much more so for any other individual, whose body is completely decomposed (that he will also need anointment).

Sources: Bamidbar Rabba, Parsha 14. Toras Kohanim, Tzav, Chapter 18. Commentary of Nachmanides to Sefer Hamitzvos, root 3. Sod Yesharim by Ben Ish Chai, vol. 2, chapter 2.