It matters at UMass Amherst.

Effective immediately, faculty are prohibited from entering into a sexual relationship with any student or postdoc for whom the faculty member has any responsibility for supervision, evaluation, grading, advising, employment, or other instructional or supervisory activity.

https://www.umass.edu/newsoffice/article/new-policy-governs-consensual

The accusation doesn't say anything about whether or not Morse was the Lecturer for anyone he had relations with.

On the more speculative side, I'd imagine they would state that fact if he actually had been, but since they didn't this story seems more like a smear campaign with a touch of gay panic than anything of substance.