The barbarian* can choose which die to add.
(PHB, page 146 (emphasis mine))
The Weapons table shows the most common weapons used in the worlds of d&d, their price and weight, the damage they deal when they hit, and any special properties they possess.
Following that, a regular longsword shown on the equipments table deals 1d8 damage when they hit. The weapon die is 1d8 slashing. The Flame Tongue, when activated deals 1d8 slashing + 2d6 fire damage when it hits. So the weapon die for an activated Flame Tongue is 1d8 slashing + 2d6 fire.
Further assisted by the wording of Flame Tongue (emphasis mine):
While the sword is ablaze, it deals an extra 2d6 fire damage to any target it hits.
Note it is not an additional effect that is dealing the extra damage but the sword itself.
That is the same wording as the quote from Weapons (PHB 146). Saying "The sword deals 1d8 + 2d6 when it hits." means that it's the weapon die.
The Brutal Criticals feature specifies one die instead of saying roll weapon die one additional time, or something similar, that eliminates rolling one of each type. So a barbarian that crits with an activated Flame Tongue can choose either 1d8 slashing or 1d6 fire to add as extra damage from Brutal Criticals.
*: Blake Steel pointed out the barbarian may not be the one choosing which die Brutal Critical adds.
[...]do you have any evidence to back up that they can choose which damage dice...
Brutal Critical says:
You can roll one additional weapon damage die when determining the extra damage for a critical hit with a melee Attack.
There is no RAW about what happens if said weapon has more than one type of die. Since someone has to choose which die is added, in the absence of anything contradicting we can say the barbarian can choose the effects of his own feature.