As it turns out there are some swords that align with GRRM's vision of it from the books.
Robb Stark's Sword
GRRM mentions this on his blog on this post:
The Sword of Robb Stark is never described in any detail in my novels... but if I ever had paid it any mind, it probably would have looked quite a lot like the one seen in the show, which Valyrian Steel has replicated. So this one can almost be considered "novel authentic" as well as "show authentic."
This was a comment on the sword replica on Valyrian Steel though the replica is faithful to the one used in the show:
Though not exactly the same when you look at the replicas the sword keeps the main spirit alive so I'd say it is faithful.
The pommel was a hunk of pale stone weighted with lead to balance the long blade. It had been carved into the likeness of a snarling wolf's head, with chips of garnet set into the eyes. The grip was virgin leather, soft and black, as yet unstained by sweat or blood. The blade itself was a good half foot longer than those Jon was used to, tapered to thrust as well as slash, with three fullers deeply incised in the metal. Where Ice was a true two-handed greatsword, this was a hand-and-a-halfer, sometimes named a "bastard sword." Yet the wolf sword actually seemed lighter than the blades he had wielded before. When Jon turned it sideways, he could see the ripples in the dark steel where the metal had been folded back on itself again and again. "This is Valyrian steel, my lord," he said wonderingly. His father had let him handle Ice often enough; he knew the look, the feel.
A Game of Thrones, Jon VIII
Like Longclaw this isn't an exact copy but it does seem similar enough that they can be considered to be faithful to the books.
"Because you are so beautiful." His hands were large and strong, and there was something in his hard blue eyes and great curving nose that suggested the fierceness of some splendid bird of prey. "Prendahl talked too much and said too little." His garb, rich as it was, had seen hard wear; salt stains patterned his boots, the enamel of his nails was chipped, his lace was soiled by sweat, and she could see where the end of his cloak was fraying. "And Sallor picked his nose as if his snot was gold." He stood with his hands crossed at the wrists, his palms resting on the pommels of his blades; a curving Dothraki arakh on his left hip, a Myrish stiletto on his right. Their hilts were a matched pair of golden women, naked and wanton.
A Storm of Swords, Daenerys IV
Shagga Son of Dolf's Battle Axes
This one is actually quite straight forward, both quotes from the books and show appear to come to the same conclusion on Shagga's weapons.
Lord Lefford frowned. "I saw that great hairy one today, the one who insisted that he must have two battle-axes, the heavy black steel ones with twin crescent blades."
"Shagga likes to kill with either hand," Tyrion said as a trencher of steaming pork was laid in front of him.
A Game of Thrones, Tyrion VIII
The show also uses a similar quote:
Kevan Lannister: "The great hairy one insisted he must have two battle-axes, heavy black steel, double-sided."
Tyrion Lannister: "Shagga likes axes."
Game of Thrones, Season 1 Episode 9, "Baelor"
Swords that aren't faithful
For completeness sake I am also going to cover some of the main swords in the series. These are ones that are named and shown or belong to a main character (also mostly ones that Valyrian Steel have also created replicas for).
As a general comment I just want to repeat that Valyrian Steel is:
Your source of officially licensed replica weapons and armor from HBO® 's hit TV series Game of Thrones® , and George R.R. Martin's A Song of Ice and Fire book series.
As such their depictions of the weapons should be pretty accurate, especially since apparently they work with GRRM to design them:
We manufacture weapon and armor replicas from George R.R. Martin's bestselling book series. Each item is an individually serialized limited edition and comes with a certificate of authenticity personally autographed by Mr. Martin himself. Each item is designed while working closely with Mr. Martin to make sure they fit his vision.
Also that their A Song of Ice and Fire swords come with the following notices, the following is for needle:
This is the exclusive officially licensed Needle, Sword of Arya Stark, from "A Song of Ice and Fire." All swords come with a certificate of authenticity personally autographed by George R.R. Martin. Each sword is a limited edition of 2500 pieces and the back of the blade has been laser etched in very fine print with the edition number and a copyright notice so you know it is authentic.
As such it appears that GRRM agrees that these are the versions of the swords that are true to the books.
Each of the following blades are different in the show compared to the books:
Lord Eddard Stark dismounted and his ward Theon Greyjoy brought forth the sword. "Ice," that sword was called. It was as wide across as a man's hand, and taller even than Robb. The blade was Valyrian steel, spell-forged and dark as smoke. Nothing held an edge like Valyrian steel.
A Game of Thrones, Bran I
As you can see from the above pictures the hilt is completely different. The book version has a white grip with gold and silver decoration in the middle, it also appears to be made of suede. The crossguard is quite short to the sword width of has the words "Winter is Coming" written across it, lastly, it has a folded, patterned blade characteristic of Valyrian Steel. In the show however, the hilt has a large plain crossguard and a large brown grip. This grip also appears to be made of leather.
His father glanced up. "I did. Come have a look at this." A bundle of oilcloth lay on the table between them, and Lord Tywin had a longsword in his hand. "A wedding gift for Joffrey," he told Tyrion. The light streaming through the diamond-shaped panes of glass made the blade shimmer black and red as Lord Tywin turned it to inspect the edge, while the pommel and crossguard flamed gold. "With this fool's jabber of Stannis and his magic sword, it seemed to me that we had best give Joffrey something extraordinary as well. A king should bear a kingly weapon."
A Storm of Swords, Tyrion IV
There isn't a replica for the book version of Widows Wail yet but we can expect it to be different as the blade is meant to be red and the show blade is not.
"A crimson sword might flash prettily in the sun, but if truth be told I like these colors better," said Tyrion. "They have an ominous beauty . . . and they make this blade unique. There is no other sword like it in all the world, I should think."
"There is one." The armorer bent over the table and unfolded the bundle of oilcloth, to reveal a second longsword.
Tyrion put down Joffrey's sword and took up the other. If not twins, the two were at least close cousins. This one was thicker and heavier, a half-inch wider and three inches longer, but they shared the same fine clean lines and the same distinctive color, the ripples of blood and night. Three fullers, deeply incised, ran down the second blade from hilt to point; the king's sword had only two. Joff's hilt was a good deal more ornate, the arms of its crossguard done as lions' paws with ruby claws unsheathed, but both swords had grips of finely tooled red leather and gold lions' heads for pommels.
A Storm of Swords, Tyrion IV
Again like Widows Wail it isn't faithful as it doesn't have the characteristic red blade from the books. Looking at the hilts, they too are completely different. In the show it is one piece of a golden lion and quite decorative. Whereas, in the books it has a simple gold crossguard with a lion on each end, a red grip and gold detailed pommel.
Valrian Steel Dagger
The blade Joff chose was nice and plain. No goldwork, no jewels in the hilt, no silver inlay on the blade. King Robert never wore it, had likely forgotten he owned it. Yet the Valyrian steel was deadly sharp . . . sharp enough to slice through skin, flesh, and muscle in one quick stroke. I am no stranger to Valyrian steel. But he had been, hadn't he? Else he would never have been so foolish as to pick Littlefinger's knife.
A Storm of Swords, Tyrion VIII
There is no replica of the book version yet, however, I have found this quote from GRRM claiming that they are different:
A handsome blade, I think. Though rather completely different from the dagger in the books, which was far plainer (though made of Valryian steel, with a dragonbone grip).
GRRM: Not A Blog - Swords for Sale
Dawn - The Sword of Morning's Sword
Ser Arthur Dayne, the Sword of the Morning, had a sad smile on his lips. The hilt of the greatsword Dawn poked up over his right shoulder.
"And now it begins," said Ser Arthur Dayne, the Sword of the Morning. He unsheathed Dawn and held it with both hands. The blade was pale as milkglass, alive with light.
A Game of Thrones, Eddard X
Although there isn't a replica of the book version, in the show Ser Arthur Dayne dual wields two swords so it's possible they changed the sword into two. However, When Ned rides upto the Kingsguard Arthur has a sword out in front of him. This looks to be Dawn.
If it is then the shows version doesn't look to be faithful to the books as it doesn't look to be a greatsword and does not have a pale milkglass blade.
Jaime Lannister's Sword
Although there is a lack of information in the books about his sword it appears as though it isn't faithful.
Jaime Lannister poked at Ned's chest with the gilded sword that had sipped the blood of the last of the Dragonkings. "Would she? The noble Catelyn Tully of Riverrun murder a hostage? I think … not." He sighed. "But I am not willing to chance my brother's life on a woman's honor." Jaime slid the golden sword into its sheath.
A Game of Thrones, Eddard IX
It isn't totally clear and open to interpretation but knowing that Oathkeeper and Widows Wail are both red I'd assume that it means for the blade to have been golden. Also it is mentioned as being golden on numerous occasions leading more credit to it actually having a golden blade.
The polished skulls of the last dragons staring down sightlessly from the walls of the throne room while the Kingslayer opened Father's throat with a golden sword.
A Game of Thrones, Daenerys I
As such the show version doesn't seem to match this description.
Swords where it's unclear
The following swords are unclear as to whether or not they are faithful.
Khal Drogo's Arakh
There isn't much description on Drogo's weapons, the only bit I could find was his "brides gifts":
Haggo gave her a great leather whip with a silver handle, Cohollo a magnificent arakh chased in gold, and Qotho a double-curved dragonbone bow taller than she was.
A Game of Thrones, Daenerys II
As it's unclear whether or not this was the Khal's arakh of choice I can't say for sure whether the show was faithful. However, the show replica is not to that description.
There is little to no description of the sword in the books other than it's a Valyrian steel greatsword.
Drumm nodded, one hand on the hilt of his Red Rain. "Lord Tarly bears the greatsword Heartsbane, forged of Valyrian steel, and he is always in Lord Tyrell's van."
A Feast for Crows, The Reaver
As such it's unclear whether of not the shows version is faithful to the books.
King Robert Baratheon's War Hammer
This is the odd one of the bunch in that it isn't technically a sword but I think it still fits within the scope of the question. It's also the only one where it is clear from the books perspective of what it should look like but not the show. The following image is of the books replica: