The Torah speaks of eyes from the very start of Chumash as the organ of desire and temptation: Eve saw that the tree was a delight to the eyes (Bereishit 3:6), and the mitzvah of tzitzit exists precisely so that the eyes and heart not be followed into straying (Bamidbar 15:39), a point Rashi (Bamidbar 15:39) sharpens by calling the heart and eyes the body's agents for sin, while the Ramban (Bamidbar 15:39) reads 'after your eyes' as referring specifically to illicit desire.
Physical sight and its failing recur throughout the patriarchal narratives — Isaac's eyes grew too dim to see (Bereishit 27:1), Jacob's eyes were heavy with age (Bereishit 48:10), and Hagar names God 'El-roi' after being seen by Him (Bereishit 16:13) — while God Himself identifies as the maker of the seeing and the blind (Shemot 4:11, Mishlei 20:12).
Eyes also function as the measure contrasted with the heart: God tells Shmuel that man sees with the eyes but God sees into the heart (Shmuel I 16:7), bribery is condemned because it blinds the eyes of the wise (Devarim 16:19), and the Psalmist prays for his eyes to be opened to perceive the wonders of Torah (Tehillim 119:18).
Chazal develop the danger of sight as the gateway to the evil inclination: the Gemara teaches that the evil inclination controls only what the eyes see, raised in discussing concern for the priests who viewed the sotah (Sotah 8a), and elsewhere the Gemara on dining discusses how the sight of food (and a parallel teaching about the sight of a woman) enhances enjoyment, drawing on the pasuk 'better the sight of the eyes than the wandering of the soul' (Yoma 74b). Mesillat Yesharim (11:16-17) applies this to Iyov's self-testimony that his heart followed his eyes, teaching that the heart should rule the eyes rather than be drawn after them.
Midrashic and mystical sources treat the eye itself as a wonder and a limit: the Tanchuma notes that a person cannot even see what is inside his own eye, using this to humble one who seeks to grasp God's ways (Midrash Tanchuma, Tetzaveh 6:6), the Sefat Emet links the seeing of tzitzit to beholding the Divine Presence (Sefat Emet, Shlach 2:3), and the Ramchal connects Dovid's prayer to open his eyes with the deepest breach of the soul occurring through the eyes (Ramchal).
Kabbalistic sources extend the eye further into the structure of the sefirot and the Shechinah: the Shenei Luchot HaBerit explains that terms like 'eye' applied to God point to inner spiritual forces called the seven eyes of God rather than any physical form (Shenei Luchot HaBerit, Toldot Adam, Bayit Ne'eman:7), the Adir Bamarom and the commentary on the Idra Rabba describe sight as a general attachment of the soul distinct from the other senses (Adir Bamarom, Beur Idra Rabba 88a, Sulam on Zohar, Idra Rabba 57), the Zohar on Yitro details the four colors found within the eye (Sulam on Zohar, Yitro 86), the Sefer HaBiurim discusses the lights that emerge through the 'orifices of the eyes' in Lurianic structure (Sefer HaBiurim II 8:1:2), and the Kaf HaChaim records a kavanah to look at the seven letters zayin corresponding to the seven eyes of God before donning tefillin shel rosh (Kaf HaChaim (Palagi) 10:17). A related Chassidic reading interprets 'our eyes shine' as referring to the receptive sight of the Shechinah, dependent on receiving light from above just as the physical eye cannot see without light (Peirush HaMilos, ועינינו מאירות).
Ibn Ezra reads 'eyes' in Mishlei as a figure of speech for will and desire, since the Torah speaks in the language of men (Ibn Ezra on Mishlei 22:12).
Halachic literature also treats the physical eye directly: the principle of 'an eye for an eye' is explained by the Oral Torah as establishing monetary compensation rather than literal retribution (Peninei Halakhah, Likkutim II 16:4), the Mishnah and Gemara dispute whether applying blue coloring around the eyes on Shabbat is a biblical or rabbinic violation (Gray Matter III, Cosmetics and Halacha), and the Admat Kodesh discusses the permissibility of preparing eye remedies on Shabbat due to the danger of eye ailments (Admat Kodesh, Orach Chayim 6:2).
On the practical question of eye transplantation, Tzitz Eliezer (14:84) addresses the halachic issues directly, generally prohibiting removal of an eye from a Jewish corpse but permitting transplantation in cases of uncertainty or serious danger, allowing an operating physician to assist under specified conditions, and requiring unused tissue to be buried rather than sent for pathological examination.