Which major magic-related Game of Thrones book characters have been missing from the HBO series from S3-S5?

Question

Which major characters with dialogue, inner or outer, who are important to the magical aspects of the story are missing from the HBO series so far?

Answer 1

I'm going to attempt this task by breaking down the most prominent magic-related storylines. Comments welcome where I've missed anyone. The question is S3-S5 so I've added Season 6 info in hidden spoiler tags only.

The citadel

A mysterious location, not yet (season 5) featured in the TV show, but from the trajectory of Sam's story it looks like it could feature in season 6. Includes a particularly important-sounding sub plot around glass candles, which appears relevant to the "when and why did magic come back" ongoing mystery.

• Marwyn the mage - mysterious, controversial arch-maester, seems to know a lot about magic coming back.
• Alleras - smart, talented novice, theorised to be Sarella Sand in disguise. No clues yet as to why she might be in disguise (maybe she's simply enjoying studying?) - but Alleras/Sarella appears to be an ally of Marwyn, and hints that he/she might be able to see the future in glass candles
• Pate - murdered novice, believed to be latest identity-theft victim of Jaqen or another faceless man. Has a key allowing access to much of the citadel.

Various other minor characters feature who have dialogue, but don't appear to be central to magic-related stories (e.g. the lazy novice Leo Tyrell).

In Season 6:

We see the Citadel reception and library, and discover that the mechanism from the GoT intro sequence appears to be hanging from the ceiling, but the magic-related elements of this story and characters from the books aren't (yet?) introduced.

Red priests

There's obviously something magical with these - whether it's the true religion they believe it to be is not yet established.

• Benerro - a powerful and popular red priest in Volantis who might be stoking a pro-Dany slave uprising, who impresses his followers by air-writing Valaryian using fire
• Moqorro - one of Benerro's priests. Particularly good at seeing the future, has healing powers, and appears to have some kind of agenda he doesn't share with his (often doomed) shipmates.

One anonymous evangelising red priestess in Volantis was featured in the TV show, suggesting that characters like these exist in the TV show world.

In Season 6:

No Benerro or Moqorro, but a mysterious show-only Red Priestess, Kinvara, begins prothletising in Mereen and unnerves Varys with a mysterious, seemingly supernatural level of knowledge. Also, Melisandre is surprised to discover a knack for reanimation.

Iron Islanders

• Victarion Greyjoy - initially skeptical ironborn captain who is impressed by Moqorro's powers and sails with him towards Mereen. Doesn't seem to be magical himself, but carries a magical dragon horn, is part of Moqorro's plan and seems quite keen on human sacrifices
• Euron Greyjoy - I'm not even going to try to summarise Euron in one line, let's just say "mysterious", and there are tonnes of theories about him having various powers, alter-egos, etc. Expected to feature in season 6. He has various mysterious crew, with magic-related features like shadow binder tattoos, but they have little in the way of exposition or character development so far.
• Aeron "Damphair" Greyjoy - Drowned-god worshiping zealot who likes to ritualistically drown disciples then bring them back to life: "what is dead may never die but rises again harder and stronger". It's not clear if there's anything magical here, or if it's just simple resuscitation, but he thinks it's something supernatural and holy.

In Season 6:

Aeron and Euron appear but without any apparently magic elements to their stories. Euron kills Balon in a very conventional way, and Aeron gives Euron a ritualistic drowning while crowning him but (possibly because of no inner monologue) doesn't explicitly claim it to be any kind of magic or supernatural blessing - it appears more like a test of fortitude and courage.

North of the wall

• Varamyr is a powerful warg / skinchanger
• Coldhands is... erm... Nope, no idea. Probably just a regular talking corpse riding a moose.
• Not exactly a character, but there's a subplot with Mance's army having what is claimed to be the Horn of Winter. More on horns here.

In Season 6:

Still no Varamyr. Technically no Coldhands either, but an undead Benjen makes a very Coldhands-like cameo (sans moose), helping Bran out of a pickle.

Brotherhood without banners

• Lady Stoneheart - undead vengence-seeker. If you don't know who this is, don't click that link or even hover over it to see the url...
• The Ghost of High Heart - that impossibly-old lady (probably a child of the forest), who lives on a hill and trades visions for songs

Other members of the Brotherhood don't seem magic related (except for the red priest Thoros who is in the TV show), though there's a theory about Lem's origins which I can't remember or find right now.

In Season 6:

Still no Stoneheart or Ghost of High Heart, and Dondarrion appears, still (arguably) alive. The Brotherhood have become more noticably vocal (and in a few cases, e.g. Lem, more violently fundamentalist) in their devotion for the Red God, but no extra magic beyond Beric and Dondarrion's not-dying trick.

Summary

There's something all these storylines appear to have in common in addition to being magic-related: as of the latest book (ADWD), they're slowly-developing mysteries where it's not yet clear what impact(s) they will have on the previously established main storylines.

One of the main consistent changes from books to show has been removing, delaying, or tightening up side stories. There's less screen time in the show than reading time in the books, and with a TV show you can't easily flick back to a section that didn't make sense when a clue comes up as to what it might mean, like you can with a book you read at your own pace.

With the exception of "Robert Strong"-style cliffhangers, they tend to avoid or reduce cases of having side mysteries and side characters that don't yet impact on existing major storylines in the season they're introduced. For example, instead of having a whole season with a sub plot that goes nowhere that season of Ser Dontas having clandestine meetings with Sansa, they saved re-introducing him until the same season in which he had a major impact on the plot.

We don't know for certain, but my suspicion is that not all of the above are permanently dropped from the TV show story - based on past form, I suspect some will be introduced in whatever is the season in which they will have whatever their first (as yet unknown) impact is on the main stories (like they did with Ser Dontas meeting Sansa).

In Season 6:

...it drops a few clues about directions in which storylines appear to be opening, but they're all more in the category 'taster or cliffhanger for what might happen next series' than 'resolved or developed storyline'.

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There does seem to be less magic overall in the TV show, but there are a couple of examples of magic in the TV show that doesn't (yet?) feature in the books:

Children of the forest throwing fireballs in season 4, and a "boss" white walker/other raising the dead by a gesture of his(?) arms in season 5.

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I'm sure I've forgotten something obvious, please do point out omissions in comments.