The way the locals pronounce it is /ou'leɪθʌ/, as you can hear early in the Mayor's speech. This is remarkably irregular: I can't think offhand of any other word in the English language in which a final e is pronounced /ʌ/, except occasionally "the". You might expect the etymology to provide an explanation, but it turns out to complicate the story further.
The town of Olathe was founded in 1857 by one John Barton, a doctor from Virginia, who - by his own account - got it into his head that "beautiful" would be a good name for the town he envisaged, and:
... meeting Capt. Joseph Parks, head chief of the Shawnees, he said: 'Captain, what in the Shawnee language would you call two quarters of land, all covered with wild flowers? In English we would say it was beautiful." Parks replied: "We would say it was 'Olathe,' "giving it the Indian pronunciation Olaythe, with an explosive accent on the last syllable. Barton made the same inquiry of the official interpreter, an educated Indian, who made the same reply, adding that for English use it would be best to pronounce it "Olathe," with the accent on the second syllable. So it came to pass that the new town was named "Olathe," the city beautiful. (History of Johnson County, Kansas)
In Shawnee, an Algonquian language, (h)oleθí is indeed documented as meaning "pretty" (Gatschet II:2, II:6, III:5); the root also seems to mean "good", judging from its occurrences (spelled <lafi>) in Alford's Shawnee New Testament translation, eg in Matthew 5:45, 19:16, 20:15. One might assume the Shawnees had their own name for the place, but that is not necessarily true, considering they had gotten there barely a generation earlier. Originally from Ohio, they were induced to sign a treaty to move to Kansas in 1831, onto land originally belonging to the Kaws (Kanzas). A few years after the foundation of Olathe, they were pushed out again, to Oklahoma.
It thus seems pretty clear that the original pronunciation of the town's name was /ou'leɪθi/, corresponding better with the spelling (cp. "synecdoche"). How did that turn into /ou'leɪθʌ/? I think the answer lies in English sociolinguistic variation. In the 19th century, standard English word-final /ʌ/ was often pronounced dialectally as /i/, yielding forms like "Americkee" for America or "Canadee" for Canada. In more recent times this pronuciation seems to show up mainly in caricatures of rural or Appalachian speech. The current pronunciation of Olathe as if it were Olatha can thus best be understood as a hypercorrection by people who didn't want to sound uneducated.
Update: A very helpful article linked by Y below, The Pronunciation of Missouri, reveals that the phenomenon is more systematic in the area than I had realised: it extends not only to placenames like Missouri, but even to words like spaghetti, macaroni, or prairie. This makes hypercorrection seem a less likely explanation. Instead, it looks as though final /ɪ/, which becomes /i/ in standard American English, was instead reduced to schwa in parts of the Midwest, including the area surrounding Kansas City. Andrews' (1994) Shawnee Grammar indicates that Shawnee /i/ was often realised as [ɪ], so this fits together nicely.