Has there been any examples of office of student affairs or office of Diversity Equity and Inclusion (DEI) or equivalent offices in university administrations providing (direct and nontrivial) input in the hiring, evaluation, and promotion of (academic) faculty?
There are usually rare cases of just about everything. However:
An office of diversity would most likely be involved in setting policy that other parts of the university must adhere to. But, in the US, at least, those are unlikely to affect individual cases other than at the margin. Diversity is still a difficult problem that has few answers, partly because it is contentious. (I have no experience outside the US.)
The office of student affairs will often collect complaints from students. Those that can't be handled immediately will probably be passed on to other administrators that have direct responsibility for faculty. So, enough complaints, forwarded to the department chair might well have an effect on tenure or promotion, but it would be more likely to be other than direct involvement by student affairs.
While the above is a view of what happens officially, who can say what goes on in private meetings. A faculty member with a sufficiently bad reputation that reflects badly on the university, might be handled sub-rosa.