Yes, quite a few.
Men
Beren was the only one.
None have ever come back from the mansions of the dead, save only Beren son of Barahir, whose hand had touched a Silmaril; but he never spoke afterward to mortal Men.
(The Silmarillion, Of Men)
Elves in general
'Reincarnation' or re-embodiment is the usual fate for Elves who die, or at least the Eldar. Tolkien expanded on this in great detail in "Laws and Customs of the Eldar" and "Athrabeth Finrod ah Andreth", published in Morgoth's Ring (volume 10 of The History of Middle-Earth).
The details of exactly how it works are not wholly consistent, but it seems that Tolkien eventually rejected the original idea of Elves being reborn as new children
A houseless fea that chose or was permitted to return to life re-entered the incarnate world through child-birth.
(Laws and Customs of the Eldar)
replacing it with a 're-housing' in a reconstructed body identical to the one they had at the time of death.
[Eru speaking to Manwe]
Look and ye will find that each spirit of My Children retaineth in itself the full imprint and memory of its former house
[...]
After this imprint ye may make for it again such a house in all particulars as it had ere evil befell it. Thus ye may send it back to the lands of the Living.'
(Athrabeth Finrod ah Andreth, The Converse of Manwe and Eru)
In any case, it appears to be the normal fate of Eldar who die, at least if they want to return to life:
The Eldar say that more than one re-birth is seldom recorded. But the reasons for this they do not fully know. Maybe, it is so ordered by the will of Eru; while the Re-born (they say) are stronger, having greater mastery of their bodies and being more patient of griefs. But many, doubtless, who have twice died do not wish to return.
[...]
Concerning the fate of other elves, especially of the Dark-elves who refused the summons to Aman, the Eldar know little.
(Laws and Customs of the Eldar)
Glorfindel
The unusual thing about the somewhat debatable) case of Glorfindel isn't that he was re-embodied/reincarnated, but that he returned to Middle-earth from Valinor afterwards.
They 'normally remained in Aman'. Simply because they were, when rehoused, again in actual physical bodies, and return to Middle-earth was therefore very difficult and perilous.
(Athrabeth Finrod ah Andreth, Note 3)
Miriel
An anomalous case from the Morgoth's Ring material, Miriel (Feanor's mother) returned to her original body, which (being in Valinor) had not decayed.
In Aman only was there no decay. Thus Miriel was there rehoused in her own body, as is hereafter told.
(Laws and Customs of the Eldar)
Ainur
Usually Ainur could not 'die' at all as their bodies were only temporary dress not essential to their function:
Moreover their shape comes of their knowledge of the visible World, rather than of the World itself; and they need it not, save only as we use raiment, and yet we may be naked and suffer no loss of our being.
(The Silmarillion, Ainulindale)
Gandalf
This is a special case, as he was one of the Istari (Wizards) - Maiar who were placed in "permanent" bodies and became incarnate beings like Elves and Men. Thus Gandalf could really die, as he had a real "permanent" body:
Gandalf really 'died', and was changed
[...]
I wd. venture to say that he was an incarnate 'angel'
[...]
By 'incarnate' I mean they were embodied in physical bodies capable of pain, and weariness, and of afflicting the spirit with physical fear, and of being 'killed'
(The Letters of J.R.R. Tolkien. Letter 156.)
Sauron
The final case is somewhat intermediate between the usual Ainur and the Istari. Sauron wasn't embodied as the Istari were, but he had become largely bound to his physical form via doing evil in it, so when he was killed at the Fall of Numenor and again at the Battle of Dagorlad, he had to reconstruct his body to become "effective" again.