The "CanDestroy" tag allows mapmakers to list specific blocks that they want to allow a player in adventure mode to break using a tool.

I expected that adding fire to the list would allow me to give a player in adventure mode the ability to destroy fire.

While the tool does display "Can Break: Fire" in game, actually hitting fire with the tool does nothing. It seems that there is no way to allow players in adventure mode to destroy fire without letting them destroy the blocks beneath it.

To reproduce, you will need a way to edit the NBT tags in your world (such as NBT Explorer). The simplest method is adding the tool of your choice to your inventory, then exiting your world. Open up the world in your NBT editor, then look within level.dat > data > player > inventory to find the tool (it will be a compound tag with 4-5 entries; the Slot and id tags should help you find it). Within the "tag" Compound tag (create it if it isn't there), create a List tag named CanDestroy. Within the list, you simply need to make a String tag for each block you would like the player to be able to destroy in Adventure mode (the String should hold the block's ID, such as minecraft:stone; you can add something to test it if you'd like). Add a String tag for minecraft:fire, then save, and open your world back up in Minecraft. Set your gamemode to adventure, and try to break the appropriate blocks – for any block IDs you added, the tool should mine them as if you were in survival, with the (sole, I believe) exception of fire.