If Moshiach comes during the week of Pesach Sheni, will we be able to bring the Pesach offering even if we're impure?
Let us first understand the background to this question: While in exile, all of us are presumed to be in a state of impurity, which comes from having contact with a dead body. (Even being in the same room as a corpse, or visiting a cemetery, is enough to render someone impure.)
The process of purification takes one week, and involves, among other things, being sprinkled with the ashes of the Red Heifer. In our time, when there is no Holy Temple, the Red Heifer cannot be slaughtered or burned. To become pure, one must wait a minimum of seven days after being sprinkled with the ashes. It follows, therefore, that when Moshiach comes, it will take at least a week until we will be purified.
Now, regarding the original Pesach offering, the Mishnah rules that if the entire community is impure, the sacrifice can be brought, even while impure. However, this ruling was made only for the original Pesach offering, not for the "makeup" date of Pesach Sheni.
The reason that the "exemption" does not apply to Pesach Sheni is that the makeup date was instituted only for the sake of those who were impure on the original date of Pesach, and were thus unable to bring the sacrifice. If so, it would defeat the purpose to allow the Pesach offering to be brought on Pesach Sheni while impure.
The Lubavitcher Rebbe raises a point: The reason we were unable to bring the Pesach offering on the proper date was not impurity, but rather because the Temple was not built. If the Temple is rebuilt before Pesach Sheni, it stands to reason that we will be able to bring the Pesach offering even if impure.
The Rebbe then concludes: "Aside from the fact that this ruling is relevant to practical halacha, since we anticipate Moshiach's coming every day, it is also possible that the very act of studying this ruling will have a spiritual effect, and we will be able to bring the Pesach offering in actuality."
(Sources: Pesachim 95:2. Yosef-Ometz L'Chida, Ch. 6. Likutei Sichos, Vol. 12, p. 220-. B'Tzel Hachochma, p. 192.)