The Undying Lands are a place of extreme blessings, essentially the heaven counterpart of Middle Earth. Very few are permitted to go there after an ancient rift in which some elves chose immortality, and others chose to take on a mortal lifespan. This decision of the forefathers applied to all the generations that came after them, and those who were on the path of mortality were allowed to live, love, and grow old on Middle Earth, never to see the Undying Lands again. Those who had mortality were the only ones allowed to sail across the sea into the west, and return to the pure lands with blessing. This blessing is known as the Gift of the Valar.
As is well known, this gift could be bestowed on a mortal in very rare circumstances, such as Frodo carrying the ring all the way through the dangerous lands of Mordor to Mount Doom, throwing it into the fires to destroy it for good (admittedly with a lot of assistance from Gollum) and practically saving the world from the eternal evil of Sauron’s reign. Many of the other members of the fellowship were also granted the favor of the valar and allowed to live out the rest of their days in the peace of heaven, including Samwise, Frodo’s truest and dearest friend, Legolas the elf of Mirkwood, and Gimli, the only dwarf ever recorded to be bestowed with such an honor. But what would happen if a mortal being were to try to get to the Undying Lands without having this blessing, this gift bequeathed to them?
This is a complicated question and one that has several fans in an ongoing debate. Whilst Tolkien did write a lot about and explore the concepts of the Undying Lands, especially in works like The Silmarillion, he never wrote out a case-by-case scenario rule book. The most popular theory is that a mortal would never be able to make it to this sacred place undetected in the first place.
Manwe, the creator of all things, is essentially an all-seeing, all-knowing, all-powerful god of this realm, and would be able to spot a person trying to cross the seas. Without even knowing who they were, or why they were making such a dangerous voyage, Manwe would send strong gusts of winds to guide their boat back to the shores of Middle Earth. Since Valinor is a pure and untouched place, Manwe wouldn’t risk any poison or evil being brought in from the outside world to taint or corrupt those already within.
However, if a mortal were somehow to slip past Manwe, hidden among elves and masked in the glow of their immortality, most believe it to be unlikely that they would survive long in Valinor. This is because their body and soul would not be able to find harmony within the land without the gift of the Valar. Some think that their soul would live on in the Undying Lands, growing and thriving in the holy place, but their body would quickly wither and weaken to the point where they were trapped inside themselves in eternal misery. Essentially like wraiths or the now overused zombies.
Others have predicted a more Dorian Grey scenario in which the body would stay youthful and eternally beautiful within the prosperity of the lands around them, but the souls (known as the fea in Tolkien’s legendarium) would become aged and tired, and that they would want to die before long with how heartsick they were for the familiarity of Middle Earth. In essence, they would have too much of a good thing to be able to bear it for long. Or they would die instantly, as soon as they stepped foot off the ship, unable to survive the overwhelming beauty of immortality before them.
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Since, either way, they would become a poor, pitiful creature akin to Gollum, or in worse case scenarios a resentful, bitter, and possibly dangerous being, there is no doubt that the Valinor would detect them sooner or later, and either send them away to recover back in Middle Earth where they belong, or destroy them if they were too far gone/posing as threat to the beauty and purity of the sacred realm.
There have been instances of mortals trying to take the gift of the Valar by force and enter the forbidden lands without permission (as in the case of Ar Pharazon, and it didn’t end well for him or his men!) so the Valinor are aware of this and treat it as an act of treason or a play for power by Melkor or Sauron (depending on the Age of Middle Earth it happens in). So even if someone did make it all the way there, what would happen to them once they arrived is a fate that shouldn’t be wished on anyone.