Is it obligatory to fast for a week as vowed in a dream?

The sleeping person is exempt from religious obligations until they awaken, due to the Prophet's (peace be upon him) saying: "The pen has been lifted from three: from the sleeper until he awakens..." (the Hadith). So, if you uttered a vow while asleep and did not utter it while awake, it is not binding, and you are not required to fulfill it.