How do you pronounce “太监” in Mainland China? Could someone explain how to pronounce 太监 in China and Taiwan? I've heard it be tai4 jian1 as well as seen it (in Pleco) as tai4 jian4！
When 监 is in the forth tone, it usually refers to a specific government department. For example 国子监 was the education department, 钦天监 was the calender department, 太监 was the department of 内务, i.e., affairs of the royal family. Then gradually people start to use 太监 to refer to the specific occupation, or people who belong to the department. 

Then 监's meaning as a government department is died out in modern Chinese, so people begin to map it to the ubiquitous jian1, which is technically wrong.

Also at least in Mandarin, there is a tendency that when a noun consists of two characters, the second character would be read as 轻声 or the "light tone", so what you normally hear is tai4jian0. There is also a tenancy that when a character is read as the light tone, people automatically thinks that it should be the first tone. This might also contributed to the shift from jian4 to jian1. But as NS said in the comments, it could probably be just my imagination...

jian1 might be technically wrong, but there are already enough people using it to make it an acceptable pronunciation, only that the dictionaries hasn't been updated yet.

As for you, if you are taking exams, use jian4 for now, otherwise use whatever you like.