Villagers can break fully grown beetroot like they normally can wheat/carrots/potatoes however they cannot pick up or replant the beetroot seeds like they can with the other 'plantable' foods. They also cannot pick up the beetroot food item. I have tested this with the correct 'browncoat' (farmer) villagers and also made sure their inventories were empty beforehand.

Thanks,
George. (MrFreakpear - YouTube)