The bug
If the last hit that kills a mob is dealt by fire block, lava block or fire charge, the mob does not drop cooked meat.
How to reproduce
Spawn a mob that drops meat (e.g. cow)
Throw a splash potion of poison at the mob
Poison brings down the mob to half a heart
Either put a fire block or lava block on the block the mob is standing on, or dispense a fire charge at the mob
→ Observe how the mob doesn't drop cooked meat
Linked issues
is duplicated by 7
Comments
Added dying from Lava and Fire to bug report, made a clearer title and description.
Confirmed, and noticed that still works with fire aspect/fire weapons.

Is this still a concern in the current Minecraft version? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.
Confirmed in 1.6.2
Confirmed
Confirmed for 08a

Confirmed for 11b.

Confirmed for 14w21b

Confirmed for 14w28a
Confirmed for 14w28b
confrmed for 14w32d. I think the bug is that a mob must die from actual FIRE damage in order to drop cooked meat. If the mob dies from "fiery" damage (from fire blocks, lava, and fire charges) it will drop raw meat. If you run the command /entitydata @e[type=!Player] {Fire:6000} when the mobs are at 1 hit point they do indeed drop cooked meat. If you run the command /execute @e[type=!Player] ~ ~ ~ /summon SmallFireball ~ ~ ~ {direction:[0.0,80.0,0.0]} (this command summons Fire Charges at mobs) when the mobs are at 1 hit point they drop the raw meat. The same happens when you light a fire or place a lava block under a mob that is at 1 health.

Confirmed in 1.8-pre2, same conditions as above.
confirmed for 1.8.8

Confirmed for
15w44a
In 1.8 (decompiled using MCP) the reason seems to be that for various situations the game first applies the damage and then sets the fire value.
Confirmed for 1.9-pre1
Confirmed for 16w42a
Confirmed for 18w10d

Can confirm for 1.13-pre3.
Confirmed.