[The Squatter of Amon Rûdh]

But do elves have to give up their immortality? I mean Arwen did, and some others through the ages,but does it say specifically some where that elves have to do it (if so, I'm sorry and could you please point that part out).

Elves are never obliged to give up their immortality, but they may choose to do so if they so desire. Luthien becames mortal because after her death she requests that Mandos re-unite her with Beren, and Manwe, having "sought counsel in his inmost thought, where the will of Iluvatar was revealed", offers her the choice to live in Valimar, where she could forget the griefs of her life, until the end of the world or to return to Middle-earth with Beren, but do so as a mortal and to share his fate after death.
Whilst not being obliged to make a personal request of the Valar, or to enter the Halls of Mandos, Arwen makes the same decision. She elects to share the fate of her husband so that death will not permanently separate them.

In each of these cases, immortality is something that is relinquished for love: eternal separation after death is something which the Elven-women find harder to accept than to pass beyond the circles of the world entirely, yet remain with the men that they have chosen.

For those of half-elven blood, whether to share the fate of Elves or of Men is one which they make for themselves, so that Elrond binds himself to the Firstborn, whereas his brother Elros elects the life of a Man, albeit one of greatly extended span.

Further to the point about those mortals who pass into the West: it is said in The Silmarillion:

...Manwe was grieved, seeing a cloud gather on the noon-tide of Numenor. And he sent messengers to the Dunedain, who spoke earnestly to the King, and to all who would listen, concerning the fate and fashion of the world.
'The Doom of the World,' they said, 'One alone can change who made it. And were you so to voyage that escaping all deceits and snares you came indeed to Aman, the Blessed Realm, little would it profit you. For it is not the land of Manwe that makes its people deathless, but the Deathless that dwell therein have hallowed the land; and there you would but wither and grow weary the sooner, as moths in a light too strong and steadfast.'

So that Frodo, Sam and Gimli, whilst presumably not being withered by the blinding light of Aman (they could not pass that far), would eventually die in the natural course of things, as it was in their nature to do. This has almost certainly been covered in the thread to which Legalos linked his post; and I apologise for the repetition.