Dictionary for different dialects
You may try Chinese Wiktionary and English Wiktionary, each has its own merits.
https://zh.wiktionary.org/wiki/%E4%B8%8D%E5%AE%A2%E6%B0%94
https://en.wiktionary.org/wiki/%E4%B8%8D%E5%AE%A2%E6%B0%A3#Chinese
When searching for the pronunciation of a specific dialect word (since it may not be listed in Wiktionary) you can try 现代汉语方言大词典 of 古音小镜:
http://www.guguolin.com/book_fangyandacidian.php
The standard dictionary of Mainland Mandarin, 现代汉语词典 has mobile Apps (Can I use 现代汉语词典 outside of the paper dictionary?). You can use it to check standard Mainland pronunciation. However it is not free to wield its full strength.
In fact, Mainland Mandarin and Taiwanese Mandarin differ little in pronunciation. The main pronunciation differences are in certain characters, for example the conjunction  和 hé/hàn. In your case, it is only that the tone sandhi is not marked in the dictionary that makes 不 of 不客气 seem to be bù in Mainland Mandarin. The actual pronunciation of 不客气 is bú kè qi (气 here is 5th tone according to 现代汉语词典, the 客气 entry). You can learn the tone sandhi rules, especially the 一不七八 rules here: https://en.wikipedia.org/wiki/Standard_Chinese_phonology#Tone_sandhi.
The terminological differences are more evident. 中华语文知识库 is a dictionary compiled by experts from both Mainland China and Taiwan, aimed at collecting such differences in Chinese words. You can get access to it here : http://www.chinese-linguipedia.org/.