If you place a sign on the top half of a door, as shown in Screenshot (5), and break the door with your cursor pointed at the top half of the door, the sign breaks as it should, shown in Screenshot (6).
But, if you place a sign on the top half of a door (Screenshot (7)) but break the door with your cursor aimed at the bottom half, the sign is left floating. (Screenshot (8))
The rest of the screenshots show variations of this bug. Ultimately, it seems the only way to actually make the sign break with the door is to aim the cursor at the half of the door the sign is attached to. If you break the door by destroying the block beneath it, the sign will remain floating whether on the top or bottom half of the door.
Banners and item frames have this issue as well.