Out of curiosity, I did a google.
One that seems to top some people's list of 'most rare' is the future perfect continuous passive.
The example one writer uses is:
“By March next year furniture will have been being made in this workshop for 100 years.”
I think this tense you'd only consider for rarity in writing. In conversation probably 0% chance of hearing it being used.
In actual conversation, I would guess the subjunctive mood is the least used. But that's purely my guess :)