Many of the organizations iconic to the Eberron setting will remain pretty much exactly as they are in Eberron Remixed. The ones that potentially need the most change are the magical organizations, since my magical system is so significantly different than that of standard D&D-like games. Anyway, let's go through them and I'll comment briefly on each.
The Aurum. (Not the most subtle of names, but I guess few people have the benefits of a Classical education anymore.) This one was prophetic in that it is fairly close to mimicking the actual Cabal of globalists and the Establishment, or the Cathedral if you like old-fashioned Moldbuggian terminology. There's no reason to change it other than to note the obvious: my Mror Holds (and therefore my Aurum) doesn't have any dwarfs, obviously.
The Blood of Vol. I like this mostly as is, although Vol is (was) neither elf, seraphim (who replace elves in my setting) or dragon nor any kind of hybrid like that. She was just a human person, and the Dragonmarked House of Vol was a human house back in the day.
The Chamber. I actually don't think that I'd ever use this. I prefer to keep dragons as solitary, voracious and malicious monsters, not inscrutable oracles, or whatever exactly you'd describe them as in Eberron. Assuming that I even use dragons at all, which is unlikely.
The Church of the Silver Flame. In Eberron Remixed, this is just "the Church" and there's no Silver Flame. Internal corruption is not the worst enemy of the Church; rather the apathy of the world at large, which should be their natural congregation, as well as the bitter maliciousness of those who imagine that they've faced persecution because someone in the Church dared to point out their own bad behavior. In any case, ignore everything written here and just refer to my own remixed write-up of Thrane for all of your church needs.
Cults of the Dragon Below. Not exactly, in Eberron Remixed. Given that these various cults have very little in common with each other, grouping them together doesn't really make sense, and I've never been keen on the concept of "the Dragon Below." Certainly Lovecraftian or daemon-worshipping cults exist in Eberron Remixed, but are best seen as such, not under some overarching label associated with the Underdark.
Dragonmarked Houses. After applying the changes already described on the Races page, and the changes that the rules mandate on the Magic page, little needs to be changed here. Because I greatly prefer it, I've gender-swapped most of the matriarchs and other women leaders into male ones, which seems to make more sense. Women seem in broad strokes less psychologically interested in or constituted for successful cut-throat political leadership without devolving into pettiness and self-destructing (see Kathleen Kennedy as simply one notable example.)
The Dreaming Dark. Because I'm not quite sure what to do with the quori, who seem like a weird combination of ghosts and Lovecraftian monstrosities, I'm not sure what I'll do with the Dreaming Dark. But as an organized threat coming from Riedra, it has potential mostly as written.
The Gatekeepers. This one needs little change as well; as wilderness people who have the task of preventing Lovecraftian (and other) horrors from destroying the world, they're one of many such secretive protective organizations. I do see Eberron Remixed campaigns as borrowing heavily in terms of plot ideas from a kind of fantasy X-files, after all.
The Library of Korranberg. Other than the magical studying cubicles and a female high councilor, I doubt I'll change much of this one.
The Lords of Dust. Subject to the changes that I already detailed on the Daemon Wastes page, I'm using this mostly as is.
Morgrave University. I like this as the plucky yet unscrupulous younger underdog rival to some of the greater institutions on the continent. I doubt I'll change anything about it.
Order of the Emerald Claw. I don't know that I like the typical "look" described for the Order members, especially the use of a flail, which always seemed like not really a weapon that most people used in real life. I kind of prefer giving them a falx if something distinctive needs to be used. Other than that, though—undead loving terrorists with secret agendas is absolutely awesome and requires no remixing.
The Royal Families. Subject to the changes already discussed on each nations' page, this is unchanged.
The Twelve. For obvious reasons, the only things that the Twelve can acknowledge openly is their devotion to the Dragonmarked crafts and the use of dragonshards to make wondrous artifacts. In this sense, they are a direct rival to the Arcanix of Aundair. However, just like their rival, in secret, they pursue illegal and heretical sorcery and it does so with the knowledge and even greedy anticipation of Karrnath's king, just as happens in Aundair.
Wardens of the Wood. This is less of an actual faction, and just a description of "kinda ordinary people in the Eldeen Reaches", so I've decided to do away with it. Because they don't exist as an organization, the normal rangers and sheriffs and whatnot of the Eldeen Reaches simply lack centralized organization, but given the rugged individualist icon of the region, that suits them pretty fine.
The Wayfinder Foundation. This is a very D&D-ish organization with few if any correlates from the real world, so I probably won't use it much. That said, I don't have any problem with it, and can see how it might be useful to keep it pretty much as is. The leader of it in print form is a halfling, but I don't really see this as the kind of organization that a jann would run, the jann being normally my substitutes for halflings. So, it'll be a human guy instead.