Warning! This post will discus spoilery plot points from some Final Fantasy games.
J.R.R. Tolkien looms over the entire modern Fantasy Genre, and doesn't lack influence on other genres either. However Japanese RPGs seem to be one corner where Tolkien' influence is mostly indirect, with the key middle man being Tabletop Role Playing Games like Dungeons and Dragons (which originated in my home-state of Wisconsin) and it's imitators.
With Final Fantasy in particular, what's cosmetically Tolkien like about it has decreased heavily since the original game which could have been accused of outright plagiarizing D&D. And even in that first game the race refereed to as Elves aren't like Tolkien's Elves at all, they are more like Santa's helper Elves.
Still translations of The Hobbit, LOTR and the Silmarilion into Japanese had all been done by 1982. So the potential for some direct influence isn't impossible. Generally however Fantasy games in a non RPG format (like Zelda) seem closer to Tolkien to me, maybe simply because the way RPGs use Magick doesn't exactly fit Tolkien's use of it.
With Final Fantasy, what is surprising is that the Final Fantasy game that seems the most like a story I think Tolkien would appreciate, is not one of the classic games with a Medieval High Fantasy setting, but rather Final Fantasy VII, the first futuristic FF game.
Some of that could be because of FFVII's Norse mythology influence. Midgar and Middle Earth are both names that derive from Midgard.
First of all I think of the Cetra as being the most like Tolkien's Eldar of any race appearing in Final Fantasy. For starters, Tolkien never described or depicted his Elves as having long pointy ears,. Physically they look like Humans, though often idealized ones. Their difference is all in lifespan and a more mystical connection to Arda. I don't recall if the Cetra had any longer lifespans, but essentially they look like Humans but had a mystical connection to Gaia via the Lifestream.
Aerith is Cetra on her mother's side and human via her father, which fits the pattern of all Tolkien's Half-Elven. But Tolkien's Human-Elf Maiden pairings always resulted in males, a line mingling descent from both Idril and Luthien happened because Beren and Luthien's son Dior had a daughter, Elwing.
But on a more thematic level. While I hesitate to label Tolkien completely Anti-Technology, he was very concerned about industrialization, and the fear of nature being harmed by it. So all technology in his Legendarium tends to be villainous, Sauron and Saruman were both Maia of Aule after all.
Final Fantasy VII is likewise very much about fears of technology going wrong and hurting The Earth, with all the SciFi cliches commonly use to symbolize that. Sephiroth is a Frankenstein Monster, and Shinra is an evil energy company destroying the Earth's life force for profit. And of course there is a Mad Scientist.
This aspect of FFVII is notable because later FF games particularly X are all about deconstructing and rejecting the technology is bad for nature attitude. In Final Fantasy X the belief that the evil monster tormenting them is a judgment for their loosing touch with nature by letting technology get out of control is commonplace. But in fact it's an evil corrupt hypocritical religion (not as Christian as you might assume) advancing that viewpoint. And in the end they use Technology (Cid's Airship) to help destroy the beast permanently.
One of the reasons I think the story of Final Fantasy X would appeal to Gene Roddenberry. Another would be destroying a God figure who turns out to be pathetic. So Final Fantasy VII is for Tolkien fans and X for Star Trek fans (II and XII are for Star Wars and Indiana Jones fans and Tactics is for Game of Thrones fans).
Also the fact that Sephiroth's plan is to sort of merge with the Earth is arguably reminiscent of the premise of Morgoth's Ring.
I want to talk about FFVII's ending, so consider this paragraph another spoiler warning.
That ending where the Meteror is still coming even after they killed Sephiroth fully activating the Holy spell and everything seems hopeless....... then Aerith and the Lifestream suddenly show up to save the day....... is exactly the kind of moment Tolkien coined the term Eucatastrophe for.
Do other FF games have moments that could qualify as a Eucatastrophe? Eucatastrophes don't lend themselves to Video Games well because we want to feel like the victory is because of the player's work. In the entire Zelda franchise the only example I can think of is in Wind Waker, when Ganondorf is about to touch The Triforce but somehow out of nowhere The King beats him to it. (The Rito showing up at the Forsaken Fortress also makes me think of the Eagles).
The only other FF example that even comes close is the end of XIII, and I'm familiar with the endings of I, II, Legend, Mystic Quest, VI, X, X-2, Tactics and XII.