Those of Mortal Descent in the Undying Lands

[Legolas]

A final point is in the appendice the Tale of Aragorn and Arwen. At the end of his life Aragorn told Arwen to leave for the Undying Lands now that would soon be dead. It would be safe to assume this is because he wanted her to take back her immortality.

This is because Aragorn did not fully understand the situation - Arwen was a half-elf, and was given the right to choose her fate. When she chose to be mortal when her father left for Aman, she was making a permanent choice. There was no turning back for her.

As Fingolfin II has said, I've shown Tolkien's explanations of what happens to Frodo, Sam, Bilbo, and Gimli - they all die eventually. Aman is place for Frodo, Sam, and Bilbo to find peace in their life without the Ring, and finally rest. The quotes you've read about exceptions - Tuor and Luthien - are correct, but, as stated in some of those quotes, these are the only cases where the fundamental race/fate of a person is changed.

You can read that article here with the quotes from letters 154, 246, and 325.