It happens many times that when I'm in washroom, I hear Azan. I want to know if I should answer the Azan while dealing with disposal needs in a washroom as we do under normal conditions? As turning the tap on makes enough noise to block the Azan from our ears, would it be sinful to turn the tap on?
As far as I know, you are not suppose to answer the Azan loudly while you are in the washroom or in a place where people relieve themselves. According to this answer to this similar question
It is disliked to repeat the Azan while in the toilet according to the well-known opinion of the four Madhabs, because it contains remembrance of Allah. but it is permissible to say it internally (in one’s heart), and when > a person finishes relieving himself then he can say it with his tongue (outside the toilet).
Why you should not answer it verbally in bathroom?.
Because bathroom is not a clean place. There are certain etiquittes in place for reading Quran or remembering Allah. You can not recite Quran in a bathroom or when you are not clean (or when you are cleaning yourself, I should add). Therefore it is not OK to answer Azan loudly inside the toilet. I think answering it silently is totally acceptable.