Lots of published novelists make broad claims about the way fiction should be (eg. all villains should have at least one redeeming trait), realize that Lord of the Rings doesn't meet that criterion, and then make up some stupid exception for it (eg. "but that was then, and this is now" - I've seen that one verbatim before). I'm sick of the near-unanimous praise for this story, and you're about to hear all of the reasons Lord of the Rings is trash.
Boromir was right when he said, "Give Gondor the weapon of the enemy." Power is not evil and goodguys need to use it to beat people who are. I know the one ring is a special case because it causally corrupts you, but that's completely incoherent because it contradicts free will. While you could argue the ring only simulates this effect by subjecting the wearer to extreme temptation, that's not how the story treats it, and if it were there still should have been a question of trying to resist it instead of sacrificing millions of lives to win the war the hard way. Especially since Frodo slipped on the ring at at least one point during the story and his behavior wasn't even noticeably affected.
The great misrepresentation of evil. All of the forces of evil in Lord of the Rings are entities who simply seek destruction for no apparent motive; Sauron himself doesn't even get any lines. But perhaps even more harmful is that evil is portrayed as something outside of human society and that wants said society destroyed/enslaved. Real evil comes - it can only come - from inside human beings, most of whom seem normal and nice on the outside. Portraying evil the way LOTR does is extremely harmful because it makes us feel that all humans are basically good and the only evil we have to worry about is external, insentient forces trying to destroy us. Precisely the opposite is true.
It might be surprising that I listed this point below the second one when it's clearly more important. I did that because glorification of monarchy is such a common thing in fantasy that I think each individual instance of it doesn't do that much damage any more; the average reader seems to just tune it out.
Lord of the Rings is racist. I don't mean against real-life minority races, I mean against orcs and uruks and goblins and to some extent against dwarves and humans as well. There are several scenes where the three 'evil' races are shown talking, which presumably means they're sapient, and yet the story portrays these races as inherently evil. Out of an entire race of sapient beings, there would and should be at least some good orcs, uruks, and goblins in the world. It happens in the opposite direction too: can you name a single bad elf in the story? I can't. Also, it's a major plot point that hobbits are capable of resisting the Ring's corruption in a way humans cannot be. This kind of relationship between race and morality is impossible because our moral inclinations have nothing to do with our biology, and ignoring that truth can only cultivate the same feelings of group identity and stereotyping that give rise to real-life bigotry.
The magic is a giant black box of Deus Ex Machinas. We have no idea what Gandalf and Saruman can and can't do except for the things we see them do, which isn't actually very much for how powerful we're lead to believe they are. Everything they do with it hit me as a huge "What?!? They can do that? How does that work?" Cases in point: Gandalf's resurrection, the Witch-King's breaking of Gandalf's staff, Gandalf's breaking of Saruman's staff...
All of the characters are disinteresting, having unremarkable personalities and little or no sympathy factor. Many of them are defined as characters almost as much by their races and combat abilities as by their personalities.
There's no justification for the near-complete absence of women in the story. Tolkien was obviously held back by beliefs about gender roles either from feeling like he could write a woman well or from feeling like any belonged in the story. Arguably the ten most relevant characters are all male.
Also, despite the unreasonably male-dominant cost, Tolkien throws in one of the dumbest misandrist cliches as well: the scene where Eowyn responds to the Witch-King's boast "No man can kill me" with "I am no man" before killing him. Goddammit Tolkien, did we really need that?