The flame arrows spell enchants a quiver so that every arrow/bolt fired from that quiver deals an extra 1d6 fire damage. I assumed that any creature (including the caster's allies) can draw and fire these arrows.
The caster has the Flames of Phlegethos tiefling racial feat (Xanathar's Guide to Everything, p. 74-75), which has the following benefit, among others:
When you roll fire damage for a spell you cast, you can reroll any roll of 1 on the fire damage dice, but you must use the new roll, even if it is another 1.
It is obvious that if the caster attacks using the flame arrow (an arrow from the enchanted quiver), Flames of Phlegethos would work for them because they are the one cast the spell. However, how about the other creature firing the flame arrow? If they are not the one who cast the spell, can they reroll the extra fire damage on a 1?