"My uncle does not like talking about taxes."
Translation:Mój wujek nie lubi rozmawiać o podatkach.
My understanding is that "mówić" means more along the lines of "to speak," and "rozmawiać" is more like "to converse." In this case, why isn't "mówić" accepted?
Well, your understanding is good, that's for sure. I personally wouldn't use 'mówić' here, but I also wouldn't consider it wrong, just a lot less probable. Why? When you use "rozmawiać", one can understand that your uncle doesn't want to get into any political, controversial issues and argue with people. In my opinion, "mówić" in this context would be going more into a direction of 'telling a story':
-Wujku, wujku, opowiedz mi historię o podatkach!
-Nie Aniu, nie lubię mówić o podatkach.
If I see "nie lubi mówić o", I wonder why he doesn't want to talk about the topic. I imagine sentences like "Ona nie lubi mówić o swoim synu" (because her son has died and it is too painful to talk about him), or "On nie lubi mówić o tamtym dniu" (because what happened on 'that day' makes him too ashamed), or even simple "Ona nie lubi mówić o sobie" (because she's a modest person).
Am I making any sense? If I'm taking my interpretation too far I'm sorry, I just woke up and my brain is not that fresh yet ;)