Should a Bugbear PC take damage when holding an enemy on the other side of a Wall of Fire with Grapple?

Bugbears have long arms, allowing them to grapple enemies 10 ft. away.

The Bugbear’s square is not in the effect range of the Wall of Fire. However, about 5 ft. worth of one of the Bugbear’s hands is supposedly in a square that is.

Mechanically, should the Bugbear take damage? If so, simulationally, how do we reconcile that such a small fraction of the player being in the square and taking just as much damage as a character residing fully in the area of effect?