I know that, in the Lord of the Rings and associated works, Eru Ilúvatar is basically the Christian God, with Morgoth being Satan. Since Tolkien famously hated allegory, they should be understood as literally being those figures, not as metaphor for such.
Tolkien occasionally hinted at or stated Christian (specifically Catholic) doctrine, for example in the need for divine intervention to save Frodo from the temptation of the Ring, or the statement that Eru might have to personally redeem humanity at a later date. One might imagine that this extends to the afterlife.
We know a little bit about what happens to creatures that die — Elves are bound to Arda until its end, say — but all I’m aware of for Men is that they “pass out of Arda”.
However, did Tolkien ever state or imply that a literal Christian afterlife existed for those humans who died in Arda? In particular, is there any mention of Heaven, or (even more so) Hell? Conversely, did he imply that a different situation held (at least at that point in time)?