Does expiation become obligatory every time the sin is committed without reciting the Quran after swearing an oath on the Mushaf to do so, or is the oath nullified by a single expiation after the first time? And what is the expiation?

The phrasing you used indicates repetition, and you are obligated to offer expiation every time you repeat the violation of the oath. The expiation is: feeding ten poor persons, or clothing them, or emancipating a slave. If you are unable to do so, then you must fast for three consecutive days, as a matter of precaution.