A face can't hide and a heart can't know; also a heart can't be false (in the sense 'duplicitous'); nor is the heart the seat of motivation. And the face isn't the only thing that's involved in giving an inaccurate impression. So yes they are both being personified (among other metaphors). It's saying 'that which has been done by a person with duplicitous motives...' (what false heart doth know) '...must be hidden by dissembling behaviour' (False face must hide). But the original 9 syllables are infinitely more telling than my multi-syllabic 'translation', don't you think? ;-)
There is alliteration between 'false' and 'face'; there is assonance between 'doth' and 'must ' (/ʌ/) - although whether there was when Shakespeare was writing, I don't know.