[Raynor]

What happens to them after the End not even the Valar seem to know.

We have this two bits:

Originally Posted by Of the severance of marriage, Laws and customs of the Eldar, The Later Quenta Silmarillion, HoME X

Since the Elves (and Men) were made for Arda, the satisfaction of their nature will require Arda (without the malice of the Marrer): therefore before the Ending the Marring will be wholly undone or healed (or absorbed into good, beauty, and joy). In that region of Time and Place the Elves will dwell as their home, but not be confined to it. But no blessed spirits from what is still to us the future can intrude into our own periods of Time. For to contemplate the Tale of Arda the Blessed must (in spirit or whole being) leave the Time of Arda. But others use another analogy, saying that there will indeed be a New Arda, rebuilt from the beginning without Malice, and that the Elves will take part in this from the beginning. It will be in Ea, say they - for they hold that all Creation of any sort must be in Ea, proceeding from Eru in the same way, and therefore being of the same Order. They do not believe in contemporaneous non-contiguous worlds except as an amusing fantasy of the mind. They are (say they) either altogether unknowable, even as to whether they are or are not, or else if there are any intersections (however rare) they are only provinces of one Ea.

Originally Posted by Notes on motives in the Silmarillion, Myths Transformed, HoME X

Men will also 'fade', if it proves to be the plan that things shall still go on, when they have completed their function. But even the Elves had the notion that this would not be so: that the end of Men would somehow be bound up with the end of history, or as they called it 'Arda Marred' (Arda Sahta), and the achievement of 'Arda Healed' (Arda Envinyanta). (They do not seem to have been clear or precise - how should they be! - whether Arda Envinyanta was a permanent state of achievement, which could therefore only be enjoyed 'outside Time', as it were: surveying the Tale as an englobed whole; or a state of unmarred bliss within Time and in a 'place' that was in some sense a lineal and historical descent of our world or 'Arda Marred'. They seem often to have meant both. 'Arda Unmarred' did not actually exist, but remained in thought - Arda without Melkor, or rather without the effects of his becoming evil; but is the source from which all ideas of order and perfection are derived. 'Arda Healed' is thus both the completion of the 'Tale of Arda' which has taken up all the deeds of Melkor, but must according to the promise of Ilúvatar be seen to be good; and also a state of redress and bliss beyond the 'circles of the world'.)