Before providing a Jewish response, it should be noted that the question posed is not identical to the one that was in New York Times Magazine. The enquiry in the Times was explicit about what action might be taken against the person in question, stating, “But might it sometimes be more ethical to provide a negative reference?” You do not allude to a specific action as a way to “to act against” the person, but given that it is a reference that has been requested, it is probably good for us to first consider the question in terms of either not giving a reference or acting against the person by giving a negative one.
Judaism does not shy against either rebuking someone or stating hard truths, but one must state them directly to the person in question (Maimonides, Mishnah Torah 6:6-7). If one is giving a negative reference, which is essentially a rebuke, you are required to let the person know what will be in it. Otherwise you would be taking the role of secretly delivering negative information about a person, which is really lashon harah, or tale bearing/gossip, which is forbidden [Arachim 15b]. Furthermore, it is presumptuous of someone to assume they can take it upon themselves to make the judgment about who is the best fit in a job. That is the task of the people responsible for filling the position.
As is implied by the question, most of us are inclined to say we will not give a reference rather than tell someone that we are only willing to provide a negative reference. Effectively, one could attempt to intervene in the hiring by offering to write a reference and letting the person know it will be negative. Of course this would most probably result in them withdrawing their request, which amounts to the same thing as not providing the reference. Not surprising, it is easier to choose to decline a request then mention the option of a negative one, but the later would be an appropriate way of attempting to intervene in the situation when one believes the person is not competent.
Would there ever be a circumstance where we would be justified, even obligated to inform someone that they should not hire an applicant? Yes, if a life were at stake (pekuakh nefesh) or if the person was engaged in ongoing financial impropriety that you knew about directly, but then these should have been addressed earlier at your current workplace. The question being posed is framed in terms of competence rather than issues of safety or legality so this would not seem to apply.
Many of us have watched people we judged to be unsuited for a position get it, often over candidates we respect. But if our evaluation is not being directly sought, no person or law is being threatened, and the person in question has declined our offer of a negative reference, then Jewish tradition does not permit us to actively work against them.