When I was a student, I always got nervous on the ___ of exams. (A) date (B) eve (C) time (D) paper

B is correct. Is A also acceptable?
on the date
is strange.

on the day

B's fine.
I think only 'on the day' is natural in the original context, not 'on the date'.

Several of the 'on the date of exams' googles are not genuine examples as the 'on' related to a prior word, not 'the date of exams'. For example: based on the date of exams = (based on)( the date of exams)
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Well, Google hits:
134 for "on the day of exams"
84 for "on the eve of exams"
13 for "on the date of exams"
so I guess I was right.
Students: We have free audio pronunciation exercises.
 nona the brit's reply was promoted to an answer.