The correct option is
B neither does Tom
The second half of the sentence presents an additional idea that is similar to the first half.
Option B: 'Neither does Tom' is an elliptical sentence of the form 'neither + auxiliary (does) + subject/pronoun (Tom)'. 'Neither' indicates an agreement with negative sentences.
It is used when a similar idea is shared in a previous sentence, or the first half of a sentence. (An elliptical sentence is a sentence that has information missing. That information must be provided in a previous clause or sentence.)
This option completes the given sentence in meaning, hence B is correct.
Option A: 'So' in 'so doesn't Tom' expresses an agreement with positive statements. Since the given statement is negative, Option A is incorrect.
Option C: In 'neither is Tom', 'is' is not the right auxiliary to use. 'Is' indicates a state of being, whereas the action indicated before is 'to do'. (I never do eat potatoes). The latter half must agree in terms of action with the former.
Option D: 'Neither Tom does' is not of the correct form. As stated earlier, the auxiliary comes before the subject, and not after. Hence option D is incorrect.