My reaction was similar to most when I first read the question. Thanks to the discussion above, I see the problem differently.
At first glance, answers A-D are similar enough that, when one speaks casually, it is easy to use inflection to convey enough meaning to be understood. However, since this is a test question, we must assume that the point is to consider the problem in the context of the need for precision in writing where a reader does not have the luxury of tonal cues or the ability to ask for immediate clarification.
This question challenges students to consider the nuances of the information that is being conveyed. The amount of information that each sentence communicates contains different levels of precision.
My opinion, now, is that if either answer A or B were the source question, then the other would be the best answer among the five sentences. Both answers A and B convey basically the same information strait-forwardly with no implication about why there is no deadline on Tuesday.
Answers C and D add additional information to the statement that Tuesday is not a deadline. C labels it unimportant, while D classifies it as not essential. The sentence in question, also, provides a reason for Tuesday not being a deadline: It isn't necessary.
If proper deductions are made by the test takers, then A and B should be dismissed categorically. Students are left to decide between which of unimportant or not essential is the closer match to isn't necessary.
As has been stated above, not essential implies there is something whose essence is not affected by the task being completed by Tuesday. It would not be overreaching to infer that this task is a component of something else, and
the deadline isn't necessary because it does not affect its essence.
On the other hand, unimportant does not convey any more information than isn't necessary. The two are closely matched in scale. Something can be inherently important or necessary without affecting the essence of a thing, even itself. We as the listener/reader do not know why a Tuesday deadline is unimportant or unnecessary, in both cases, we just know that it is. Therefore, for me, the answer is C and I hope for the sake of the students that this is an advanced course.