Probably, but not very much compared to the changes that happened later. The introduction of hangul probably had no significant impact on the very large number of homophones we see in modern Korean.

As an introduction to the fuller answer, the history of the Korean language is divided into Old Korean (to the end of Silla), Early Middle Korean (Goryeo), Late Middle Korean (Joseon until 1600), and modern Korean (1600 to today). We know very little about the pronunciation of Old Korean - relatively few writings have survived, and the language used for writing in Korea then was Classical Chinese. There were multiple languages in Korea in the early part of this period, with the unification of Korea by Silla leading to a Korean language (instead of many Korean languages). We know a lot more about Middle Korean, especially after the introduction of hangul (early in the Late Middle Korean period). Modern Korean underwent significant evolution, especially with the 20th century standardisations of the Korean language (similar, but not identical, in North and South).

There are two main reasons why we see so many homophones in Korean today. First, the majority of the vocabulary is derived from Chinese. Second, unlike the Chinese languages, Korean is not a tonal language - many of the words that are homophones in Korean are not homophones in Chinese languages because they are pronounced with different tones.

Back to the first part of the answer: the (probable) effects of the introduction of hangul on Korean pronunciation. Hangul was accompanied by an attempt at the standardisation of Korean pronunciation and Hangul spelling; this is suggested by the title of the book introducing hangul: Hunminjeongeum, "The Correct/Proper Sounds for the Instruction of the People" and its later, and longer, commentary: Hunminjeongeum Haerye, "Explanations and Examples of the Correct/Proper Sounds for the Instruction of the People". This resulted in the standardisation of written Korean to the end of Middle Korean. We don't know the effect of this on the spoken language, but there was probably some, but less complete.

At this point in time, there were far fewer homophones in Korean. First, less Chinese vocabulary had been absorbed, and more of the original Korean words were still in use. Second Middle Korean was a tonal language, with three tones: "even tone" (low), "departing tone" (high), and "rising tone" (rising). Tones were indicated in written Korean using one dot to the left of the syllable for high, and two dots for rising. This meant that there were some homophones in Sino-Korean words, since Middle Chinese used four tones ("level" which became low tone in Sino-Korean words, "entering" became high tone, and both "rising" and "departing" became rising tone), but far fewer than in modern Korean.

The break between Middle Korean and modern Korean is easily recognisable in literature: tone markers disappear, the hangul consonant for z disappears, and spelling varies greatly. The break happens during the Japanese invasion of Korea in 1592-1598, and the post-war literature is in the new style. This is very unlikely to be caused by such a sudden change in the Korean language, and much more likely to be caused by the dropping of old scribal conventions for writing Korean that had led it to no longer representing the spoken language. A large - and new - part of the post-war revival of Korean literature was Korean-language literature for the masses (at least, for the middle classes, and for educated women). Elite literature continued to be written in Classical Chinese, and doesn't tell us about changes in Korean language. The new vernacular literature shows us both the evolution in spoken Korean that had occurred after the introduction of Hangul during Late Middle Korean, and the evolution of early modern Korean.

The replacement of Korean words by Sino-Korean words continued, and the number of Korean homophones continued to rise. This, together with the disappearance of tones, made modern Korean very homophone rich.

In some cases (but not very many), some Sino-Korean words became conventionally written in hangul in mixed-script writing (when Korean was written in a mix of hanja and hangul). This suggests that the Chinese origins of these words was forgotten, and writers assumed that they were old Korean words. In same of these cases, the pronunciation changed over time (as did the pronunciation of many Korean words), and became different from how its former homophones were pronounced (and the hangul spelling changed to match).

A postscript: The new wave of vernacular literature was a huge addition to the former literature focussing on education of the elites and Buddhism. It included travel journals, popular poetry (written in Korean, instead of Classical Chinese), cookbooks, and novels. One of the early novels reflected the birth of the new literary era: Imjin Nok, "Record of the Black Dragon Year", on the Japanese invasion. There were also Korean translations of Chinese works. The new literature circulated in both printed and manuscript forms, often via commercial library services which meant that readers could pay much lower prices per work read.

A reference:

Ki-Moon Lee and S. Robert Ramsey, A History of the Korean Language, Cambridge University Press, 2011.

Hello! Intermediate Korean speaker here. I also love the history of languages and the Korean language has a very fascinating history! Prior to the invention of Hangul, Chinese Hanja was used as the written language of Korea. For quite some time, the Korean language was actually mildly tonal- although this is not related to the use of Hanja. In your example of the word 무, it is possible that tones at one point helped to separate them into distinct words, but I'm not sure. There is however a fantastic source here that provides us with a better understanding of Korean tonality in Middle Korean. For example, the word 손, romanized son, could mean both "guest" or "hand." Each meaning previously had it's own distinct word as sòn and són, respectively. After the invention of Hangul and King Sejong the Great's decision to not use markings to indicate tonality in it, the two words written as 손 became homonyms over time. So, the invention of Hangul absolutely affected the pronunciation of Korean over time in that it is the primary reason for the shift away from the use of tones in modern Korean.

Lee, S. (1979, February 23). On the Origin of Korean Middle Tone.