The word "who" relates to things like your name. The word "what" relates to things like your gender. If you are talking about or referring to being "an Italian" for example, (your name is Giovanni Rossi), the sentence, "I will follow my dream and be who I am," works better. If you are talking about or referring to being "gay" for example, (you identify as a homosexual), and you want to "come out of the closet," the sentence, "I will follow my dream and be what I am," works better.

Who is only used for people. The word 'what' can be used, but it is better to use it for everything except people. You are a person and so I 'be who I am' sounds best.