Isaiah chapters 56-66 were, apart from some minor insertions from other periods, written shortly after the Return from Exile by an anonymous source now known as Third Isaiah. The returning Jews were grateful to the Persians who, as foreigners, liberated them and undertook to rebuild the Jerusalem temple. This gratitude is demonstrated by the lavish praise God is described by Second Isaiah (chapters 40-55) as heaping upon the Persian king, Cyrus:
Isaiah 44:28-45:1: That saith of Cyrus, He is my shepherd, and shall perform all my pleasure: even saying to Jerusalem, Thou shalt be built; and to the temple, Thy foundation shall be laid. Thus saith the LORD to his anointed, to Cyrus, whose right hand I have holden, to subdue nations before him; and I will loose the loins of kings, to open before him the two leaved gates; and the gates shall not be shut;
Having owed so much to the Persians, it is plausible that the Jews would tolerate them in the temple or perhaps even serving as priests. Plausibility is, of course, not proof. Nathan MacDonald (Priestly Rule page 27) says there is considerable disagreement among interpreters as to whether the oracle in Isaiah 56 envisages foreigners serving as priests. He says (page 29) the passage can and has been read as a promise that foreigners will take up priestly roles, and that the earliest textual evidence is that the passage was understood in antiquity as a promise that foreigners would have a priestly role in the temple. MacDonald's own preference is to regard the intention of the writer as an expression of general devotion.
Isaiah 60:7 and 66:21 are also capable of being read as promises that foreigners would have a priestly role in the temple. If indeed these passages promise that foreigners could serve as priests, they would seem to contradict Levitical law, but the Levis were already in decline, with the house of Zadok in the ascendancy.