which is correct?

I'd guess the reason is that the word 'tall' doesn't belong to the word 'feet'.

Then neither does the 12 months "belong" to the duration. The question is not whether a possessive apostrophe is correct or not. The question is whether it is a possessive at all.
 
Agreed, you don't say "six feet's tall". English is my second language, but I'd guess the reason is that the word 'tall' doesn't belong to the word 'feet'.
Nor do we say "six feet of tall." (That's a function of "tall" not of the number+unit paradigm.) But we do say "six feet of height," even though "in" fits better. And we can get away with "a cairn of six feet's height."

The apostrophe is a replacement for "of" when it occurs after number+unit. It indicates a different kind of belonging than possession.
 
I might also sneak back in here to suggest that tall is an adjective (isn't it?). Wars have been fought but, as yet, adjectives don't have rights of possession.

Height, duration, being nouns, have the right, granted by the crown, to legally hold possessions.

It may not be fair or defensible. Take it up with adjective emancipation groups.
 
What precedes this phrase? If a preposition like “of” or “for” etc, then the noun part is “duration” and the length of that duration is the adjective and should not be possessive, so no apostrophe. It could be plural or singular depending on the preposition. “Of twelve months duration” vs “for twelve months duration”.

If the phrase is not a prepositional, then “duration” is either a subject or object of the sentence. In that case the length is an adjective of “duration”. In this case, then an article preceding the length would change things around. “It’s twelve months duration” vs “a twelve month duration”.

At least, that’s how I would approach it.
 
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