Edited by Auldrick:
When the Mob Griefing game rule is disabled, farmer villagers will not plant or harvest crops, nor pick up crop or seed items.

Steps to reproduce:

1. Enclose a flat area of up to 9 x 9 blocks using fences or a 2-high wall of blocks.
2. Create farmland with a water source within the enclosed area.
3. Ensure Mob Griefing is turned off.
4. Spawn a villager (farmer/brown coat) in the enclosure.
5. Throw wheat seeds, carrots, potatoes, or beetroots at the farmer.
   Results: The farmer should pick up the seeds or crops and plant it in the farmland, but it does not do so.
6. Plant seeds or crops in some of the farmland, then use bone meal to grow it to maturity.
   Results: The farmer should harvest the mature plants, then re-plant, but it does not do so.
7. Break some of the mature crops and leave them floating as items.
   Results: The farmer should pick up the items, but it does not do so.
8. Turn on Mob Griefing.
   Results The farmer picks up items, harvests mature crops, and plants new crops as expected.

Original description:
I have planted every type of crop and thrown every type of crop to farmer villagers and they will not plant harvest or pick up any of them.

I searched and found someone reported this saying they pick up potatoes and carrots but they do not for me so I though I'd report