The future perfect continuous is actually
He will have been eating the apples.
The apples will have been being eaten by him.
makes the sentence a passive construction: the future perfect continuous passive. You can also say 'progressive' instead of 'continuous'. This uses
will - modal
have - perfect
been - continuous or progressive
being - passive
eaten - main verb
This is grammatical, but rarely used–even in writing. The chances of someone saying it are infinitesimal. Your teacher probably demonstrated this construction because it demonstrates all the possible "characteristics" a verb might take on, including a modal.
This is a passive construction due to being. The subject apples is being acted upon by the agent him.
A sentence more likely to be encountered in authentic English does include the progressive. This is the future perfect passive:
The apples will have been eaten by him.
The apples will have been eaten by him by the time we get home.
The use of the progressive (or continuous) being stretches put the action so that it has duration and can be represented by ~~~~~~~~, but the use of the progressive in combination with the future perfect passive is rare. Few people, beyond linguistic professors, are ever going to need to express
The apples will have been being eaten by him by the time we get home.