Phrase for describing that someone is the wrong person for the job

Solution 1:

How about wrong tool for the job?

If you want to take out a screw, you choose a screwdriver, not a hammer. This doesn't mean hammers are "stupid". They're great at driving nails. They're just ill-equipped (not designed) for removing screws.

And so it is with people. You need to choose the right "tool" for the job. If you want someone to improve housekeeping, you choose a neatnik, not a slob (or so conventional wisdom would have it).

So, if Bob is the worst culprit when it comes to washing his own cup, and you make him responsible for ensuring that everyone washes their cup, I could say, with some justification, "You chose the wrong tool for the job."

Wrong tool for the job has the same connotations as "the fox guarding the henhouse": the fox is the wrong tool for the job. Perhaps this is why you thought of "the fox guarding the henhouse" when pondering your own question.

Solution 2:

Mismatch, or, responding to the OP's comment, absolute mismatch.

Bob is an absolute mismatch for the job of ensuring that all the cups are washed.

Another phrase for Bob and this task: Absolutely hopeless

We can't make Bob the clean coffee cup czar; he would be absolutely hopeless at the job.

mismatch, from Longman Dictionary of Contemporary Usage

a combination of things or people that do not work well together or are not suitable for each other

Not only can people be mismatched or things (teacups and saucers), but a person and a job can be mismatched.

Example, from Job mismatches in Pakistan: is there some wage penalty to graduates?

In this study, an attempt has been made to estimate the incidence of job mismatch and its impacts on graduate's earnings in Pakistan. The study has divided the job mismatch into three categories; qualification-job mismatch, skill mismatch and field of study and job mismatch. (Emphasis added)

As for absolute, the OED says

Complete or entire in degree; in the fullest sense. Freq. used for emphasis

Thus Bob could not be more mismatched for this particular job, if he is absolutely mismatched for it.