Did Frodo and the others (Bilbo, Sam, and Gimli) who passed over the Sea eventually die, or had they become immortal?
They remained mortal. Tolkien’s conception was that a creature’s natural lifespan was intrinsic to its spiritual and biological nature, and that this could not be altered save by a direct intervention of the Creator. There were three occasions when this did happen (Luthien, Tuor, Arwen), but it did not in the cases of Frodo & Co. Tolkien stated explicitly in more than one letter that Frodo’s journey over the Sea was only a temporary healing, and that when the time came he and the others would die of their own free will.
