The Wikipedia entry for misology provides an interesting history of the word. It notes the occurrence of the ancient Greek μισολογία in Plato's Phaedo (89d). Wikipedia also identifies Kant's use of the German "misologie" in the Grundlegung zur Metaphysik der Sitten (395.32, 1785), where Kant defines the new word as a 'hatred of reason.' According to Henry Hitchings, Samuel Taylor Coleridge based his use of "misology" on Kant's misologie.
Webster's pegs the introduction of the English "misology" as being "circa 1834." The online version of Webster's does not identify the source. I believe the author of the Webster's entry had in mind the appearance of "misology" in an article in The Quarterly Review by Henry Nelson Coleridge, nephew and son-in-law of the great romantic poet. ("The Poetical works of S. T. Coleridge," The Quarterly Review, No. 103 [1834], p. 21) The article was published anonymously in 1834, but here is a cleaner copy of it with H. N. Coleridge identified as its author.
Henry Coleridge there used "misology" while quoting his famous uncle's assessment of Goethe's epic poem: "The intended theme of the Faust is the consequences of a misology, or hatred and depreciation of knowledge, caused by an originally intense thirst for knowledge baffled." (Ibid.) Henry seems to have based the quotation on some notes of Samuel's 'table talk', since the same passage appears in the "Table Talk" entry for February 16, 1833. Apparently, S. T. Coleridge also applied "misology" to some religious sects in his book annotations.
I found an earlier use of "misology" in English. It is in John Richardson's 1819 translation of Kant's Logik (1800): Logic from the German of Emmanuel Kant. (London: Printed for W. Simpkin and R. Marshall, 1819) Here is an excerpt from Richardson's translation:
Who hates science, but does not love wisdom the less on that account, is named a misologist. Misology commonly arises from a want of scientific knowledge, and from a certain sort of vanity therewith conjoined. And sometimes those, who at first cultivated the sciences with great diligence and success, but in the end found no satisfaction in all their knowledge, fall into the fault of misology. (p. 32)I have consulted Thomas Taylor's 1793 translation of Plato's Phaedo (esp. 89d2-3), and did not find any use there of "misology." Rather than using a single word for Plato's term, Taylor uses the phrase "hatred of reason." (Four Dialogues of Plato: The Cratylus, Phædo, Parmenides and Timæus, [1793], pp. 197-8) It may be, though, that some earlier translator of Plato's work into English used "misology."