I doubt anyone shortens 아무것도 to 무것도. It is not common (except in slang and funny expressions) to chip off the beginning of a word. My guess is that you missed 아 because they said the syllable very weakly. This can happen if you slow down the beginning of a phrase putting a lot of emotion into it. Also, the beginning vowel is usually not as clearly heard as the subsequent consonant. (E.g. in 우선, 선 is always more clearly heard than 우).

There is no word 무것도 in Korean. You might have misheard something else. What is the drama you heard it?