In both the books and movies of The Lord of the Rings the elves go to the West. However it says nothing of the Mirkwood Elves heading to go West, nor does it say anything about them having any plans to.
Do they leave across the sea at The Grey Havens or do they remain? And if they do stay in Middle-earth, what happened to them? Did other elves stay as well?
I did not read The Silmarillion.
Tolkien is quite clear on a number of occasions that the last of the Eldar (i.e the Elves of the West who undertook the Great Journey) left Middle-earth in the years following the defeat of Sauron, but sometimes this may have been many years after, and he's not always clear in exactly how long.
For example, in the prologue to the Lord of the Rings we read the following:
It is said that Celeborn went to dwell there after the departure of Galadriel; but there is no record of the day when at last he sought the Grey Havens, and with him went the last living memory of the Elder Days in Middle-earth.
And in Of the Rings of Power and the Third Age (published in the Silmarillion) we read:
In that time the last of the Noldor set sail from the Havens and left Middle-earth for ever. And latest of all the Keepers of the Three Rings rode to the Sea, and Master Elrond took there the ship that Círdan had made ready... it passed into the Ancient West, and an end was come for the Eldar of story and of song.
There was however another group of Elves known as the Avari, who refused the Great Journey, and many of the Mirkwood and Lórien Elves are members of this group. It's not told whether or not these went West or remained in Middle-earth following Sauron's defeat, but one would expect that they would have little motivation for going West since they refused the summons of the Valar in the first place.
The primarily linguistic essay Quendi and Eldar, published in History of Middle-earth 11, sheds some additional light on relationships between the Eldar and the Avari, noting in particular of the Tatyarin Avari (i.e those from the second clan, from which also came the Noldor) that:
They were actually unfriendly to the Noldor, and jealous of their more exalted kin, whom they accused of arrogance. This ill-feeling descended in part from the bitterness of the Debate before the March of the Eldar began...
But also:
For in contrast the Lindarin elements in the western Avari were friendly to the Eldar, and willing to learn from them; and so close was the feeling of kinship between the remnants of the Sindar, the Nandor, and the Lindarin Avari, that later in Eriador and the Vale of Anduin they often became merged together.
So in the end the matter of whether these Elves stayed or left is left open.
To add to this discussion, I do believe Thranduil, Celeborn and Círdan all eventually left together. In addition to what has been posted here, Tolkien also writes of “The Last Ship” that sailed West in a collection of 16 poems published in 1962, which he attributes to being part of The Red Book of Westmarch. The poem speaks of a ship seen by one of Sam’s relatives sailing West with 3 elven Kings wearing crowns in the 4th age. Since we do know that Thranduil, Celeborn and Círdan all remained in Middle-earth into the 4th age, it is reasonable to conclude that this is a glimpse of their final departure. Both Thranduil and Celeborn had ruled kingdoms and Círdan had previously told Gandalf that he would remain until the final ship sailed.
There is a footnote in RotK in which Arwen ruled with Aragorn, "in glory and bliss, Queen of Elves and Men." This suggests that there were some Eldar remaining, who were subjects of Gondor and/or Arnor.
These probably left with the passing of Arwen.
