Concerning the question of eating terumah nowadays, one of the responses on the site mention a reason for not giving terumah to a cohen nowadays is because of doubts over his lineage. Yet, this would seem to be a case of ‘passeling kesheirim’. Surely, cohanim have a hazaka, besides which, scientists nowadays have found that all cohanim have the same gene. The Mashiach will not let us know, because halachah is decided here on earth, and not from the heavens! Surely, we cannot doubt the lineage of cohanim!
Cohanim today are considered such because, as Rambam (Issurei Biah 20:2) states, they have a chazakah, meaning they are assumed to be cohanim, because we have no reason to doubt their lineage.
Authorities dispute whether this implies that they are seen as vadai (certain) cohanim, or whether they remain safek (doubtful) cohanim. According to Mabit (Kiryas Sefer on Rambam) and his son Maharit (1:85), they are seen as certain cohanim. However, Rivash (94), Shach (Yoreh De’ah 322:4), Maharshal (Yam Shel Shlomo, Bava Kama 5:35), Magen Avraham (456:9; 201:2), Mahashdam (Even Ha’ezer 235), and other authorities, all write that cohanim of today are not considered “certain cohanim.”
There are many halachic ramifications of this question, which are discussed by the authorities above, such as whether we force a cohen married to a chalutzah to separate, whether we can permit a woman who was taken captive to marry a cohen, and so on.
Yet, concerning terumah, which is a particularly severe prohibition, even whose opinions who consider today’s cohanim to be certain cohanim, concede that they may not consume (Torah) terumah. For the purpose of eating terumah, the cohen’s lineage would need to be checked (Kiryas Sefer, ibid.), and for the vast majority (if not all) cohanim, this cannot be done.
In future times, there would not be a problem for Mashiach to determine the status of cohanim. Although, as the question mentions, mashiach might not be able to determine question of halachic rulings, he would certainly be able to clarify questions of factual verification, as the Gemara mentions concerning other lineage-related disqualifications.
Sources: One of the proofs for the certainty of cohanim today is the fact they recite the blessing over birkas kohanim. For one deferral of this proof, see Shevus Yaakov 1:93.