I heard a rumor that in the future era, the firstborn’s will become the new Kohanim who will serve in the temple and the Kohanim will longer serve. Is this true?

There are indeed sources which make such a claim and so was famously written by the Arizal in Sefer Hagilgulim, the Or Hachaim Hakadosh, and the Alter Rebbe who writes that not only will the priesthood return to the firstborns, but it will return to the firstborns of the mother and not the firstborns of the father, unlike the law today that we follow the father for lineage of priesthood. However, this matter is not mentioned in the Talmud or Rishonim, and is seemingly based on the debate regarding whether the Kohanim received their appointment at the time of Matan Torah, which would make it seem unchangeable in the future, being that the Torah is eternal [Opinion of Rebbe]. Or, if the appointment occurred only later prior to the building of the tabernacle, and until that point, it was indeed the firstborn’s who offered the sacrifices [Opinion of Rebbe Yehoshua Ben Karcha]. The Or Hachaim Hakadosh adds that indeed both the Bechoros and Levi’im will serve together as Kohanim.

Sources: Sources which write that will return to Bechoros: Sifri Behalosecha; Yalkut Shimoni Remez 364; Sefer Hagilgulim 23; Or Hachaim Hakadosh Bereishis 49:28; Bamidbar 3:45; Ahavas Yonason of Reb Yonason Eibashitz Haftoras Emor; Mamarei Admur Hazakein 5572 p. 128 “However, we find a clear proof of this matter from the fact that in the future, the firstborn will be Joseph, whereas now the firstborn is Reuben, not Joseph. As it is written about Joseph, “His firstborn bull,” etc., which refers to the future. For this reason, in the future, the service will be performed by the firstborns and not by the Kohanim, as it is known that the service initially belonged to the firstborns when they left Egypt, as it is written, “Sanctify for Me every firstborn, the first opening of every womb,” etc. This sanctification was for service, as it is written, “For Me, you sanctified every firstborn,” etc. However, because of the sin of the Golden Calf, the service was taken away from the firstborns and given to the Kohanim, the sons of Aaron. But in the future, the service will return to the firstborns as before. However, it will not be like the current law of firstborns, which is only to the father, just as the current Kohanim are only from the sons of Aaron according to their families and father’s house, as it is written. Thus, a Kohen whose father was a Kohen is fit for priesthood, not if his mother was a Kohenet. But in the future, the firstborns will not be associated with their father’s house to be considered firstborn only to the father, but rather to the mother, as it is written, “Sanctify for Me every firstborn, the first opening of every womb,” which refers specifically to the mother. If one is a firstborn to the mother, even if not to the father, he is fit for service. If a man has several wives, and each has a firstborn son, all are fit for service. However, in terms of inheritance, only the firstborn to the father takes a double portion, as he is considered the first of his strength and has the right of the firstborn regarding inheritance. Yet for service, it does not require being the first of his strength, but being the first opening of the womb.”; Mamarei Admur Haemtzai Devarim 2:594; Or Hatorah Mikeitz p. 686; Sources which imply that will remain with Kohanim: Yuma 5b; Taanis 17a; Sanhedrin 22b;The dispute regarding when they began the Kehuna: Zevachim 116b; Midrash Raba Naso 6; Encyclopedia Talmudit Vol. 27 Erech Kohen p. 171;