This post contains spoilers.
In re-reading The Silver Chair I found only a few things to write about (I did find an-other, but I have to do some more research on that, so I'll come back to it later).
First, there's Prince Rilian's fight with the Lady of the Green Kirtle in Underland in Chapter Twelve. She turns herself into a serpent and wraps herself around his legs, but "the Prince caught the creature's neck in his left hand, trying to squeeze it till it choked." With his other hand, he strikes the serpent with his sword, and with some help from Puddleglum and Eustace, "they hacked off its head." The narrator then mentions that "the floor, as you may imagine, was a nasty mess."
Although it's been years since I've read it, I recognized the similarities between this event and Redcrosse Knight's battle with Error in Edmund Spenser's The Faerie Queene. Redcrosse Knight descends into a hole and "he saw the ugly monster plaine, / Half like a serpent horribly displaide, / But th'other half did womans shape retaine" (canto 1, stanza 14) Like Spenser's Error, Lewis' Lady of the Green Kirtle is half woman and half serpent (although she's either/or rather than a mix). Redcrosse Knight attempts to strike her with his sword, but he misses his mark. She then "lept fierce upon his shield, and her huge traine / All suddenly about his body wound" (canto 1, stanza 18), just like the Lady does to Prince Rilian. Next, "Knitting all his force, [he] got one hand free, / Wherewith he grypt her gorge with so great paine, / That soone to loose her wicked bands did her constraine" (canto 1, stanza 19). Redcrosse Knight is able to free himself merely by choking Error where Prince Rilian still has to use his sword before he's freed, but the situation is still quite similar. In the end, just as Rilian beheads his serpent, Redcrosse Knight "strooke at her with more than manly force, / That from her body full of filthie sin / He raft her hatefull head without remorse" (canto 1, stanza 24). Finally, Error's "scattred brood... flockéd all about her bleeding wound, / And suckéd up their dying mothers blood" (canto 1, stanza 25). Spenser's description is more graphic than Lewis', but the large amount of blood is common to both.
Second, there's the temptation with which the Lady of the Green Kirtle seduced Rilian. At the end of Chapter Fifteen, Rilian recounts "the whole adventure with the older and wiser Beasts and Dwarfs" of Narnia, and "they saw how she had dug right under Narnia and was going to break out and rule it through Rilian; and how he had never dreamed that the country of which she would make him king (king in name, but really her slave) was his own country." The situation is different, but there's some resemblance between this and the devil's temptation of Jesus in Luke 4. "And the devil took him up and showed him all the kingdoms of the world in a moment of time, and said to him, 'To you I will give all this authority and their glory, for it has been delivered to me, and I give it to whom I will. If you, then, will worship me, it will all be yours'" (Luke 4:5-7). The devil tries to tempt Jesus with power over the world, but Jesus already had that power, just like Rilian already had (or would have) power over Narnia.
There's also the comparison between the Lady of the Green Kirtle and the White Witch from The Lion, the Witch, and the Wardrobe. Rilian and "the older and wiser Beasts and Dwarfs... all saw what it meant; how a wicked Witch (doubtless the same kind as that White Witch who had brought the Great Winter on Narnia long ago) had contrived the whole thing." In The Magician's Nephew, when the White Witch is still known as Jadis, she tries to tempt Digory with the fruit of the tree, just as the serpent tempts Eve in the Garden of Eden. So the comparison that Rilian and the Narnians make is more than just a comparison of devious plots; both the Lady of the Green Kirtle (literally) and the White Witch (allusively) have the guise of a serpent.