The verb after "needs" should be a gerund (verb ending in -ing) to express something required or necessary in the present. 

Option A "to cut" and B "to have cut" are infinitives which don't fit here. 

Option C "being cut" implies the action is already happening by someone else. 

Option D "to be cutting" suggests the action hasn't started yet, which isn't implied.

Therefore, the most logical answer is E. "cutting" since his hair currently requires or needs the action of being cut in the present time.