"I need to work overtime every day this week."
It depends on the context.
都 means All, and since the subject is singular "I", adding 都 is wrong in a strict sense. We can say somehow 我们明天都要加班(none of us is spared).
If the context is I work overtime today, and tomorrow it will be the same, the correct way to say it is 我明天 也 要加班. Using 都 is the way in Cantonese.