Several blocks made of wood (Signs, Ladders, Chests, Trapdoors, Doors...) won't burn

Blocks and entities made of wood that do not burn:

Affects	Block/Entity	Catch fire near lava	Burn away when on fire
Bamboo Sapling	Block
Banners	Block
Barrel	Block
Beds	Block
Campfire	Block
Cartography Table	Block
Chest	Block
Crafting Table	Block
Crimson Roots	Block
Daylight Detector	Block
Fletching Table	Block
Hanging Sign	Block
Jukebox	Block
Loom	Block
Mushroom Blocks	Block
Nether Sprouts	Block
Note Block	Block
Smithing Table	Block
Trapped Chest	Block
Warped Roots	Block
Wood Doors	Block
Wood Pressure Plates	Block
Wood Signs	Block
Wood Trapdoors	Block
Arrows	Entity
Item Frame	Entity
Paintings	Entity
Grindstone	Block
Ladder	Block
Lever	Block
Piston Arm	Block
Rails	Block
Redstone Torch	Block
Saplings	Block
Smoker	Block
Torch	Block
Tripwire Hook	Block
Wood Button	Block

The blocks that lava can catch fire to, these blocks have a flammable Material (and the other blocks have an non-flammable material).

The blocks don't burn away because they aren't listed among the burnable blocks in the BlockFire class, so fire itself isn't sustained by them, and also doesn't affect them.