"I will stay there for more than 3 weeks." Makes more sense, as the object 'there' (a location defined somewhere else in the conversation) needs a preposition to connect (the purpose of) the action. Although if this statement is directly following a sentence that defines what 'there' is ("I am going to [place]."), you could just say "I will stay for more than 3 weeks." 'Stay' in this case is associated with living somewhere.