[RPG] Does the fire damage of a bow still work with lightning arrow

From Sage Advice — Rules Answers: April 2016:

If you use Great Weapon Fighting with a feature like Divine Smite or a spell like hex, do you get to reroll any 1 or 2 you roll for the extra damage?
The Great Weapon Fighting feature—which is shared by fighters and paladins—is meant to benefit only the damage roll of the weapon used with the feature. For example, if you use a greatsword with the feature, you can reroll any 1 or 2 you roll on the weapon’s 2d6. If you’re a paladin and use Divine Smite with the greatsword, Great Weapon Fighting doesn’t let you reroll a 1 or 2 that you roll for the damage of Divine Smite.

Also: As pointed out by Doval, Jeremy Crawford allows all weapon dices to be rerolled. This includes, for instance, a Frost Brand sword's additional cold damage, since that damage is part of the weapon's damage itself (not from an additional feature).

No, you don't add those

The section of the rules on damage rolls states (PHB, p. 195):

When attacking with a weapon, you add your ability modifier—the same modifier used for the attack roll — to the damage. A spell tells you which dice to roll for damage and whether to add any modifiers.

While you are making a ranged weapon attack, the damage is that specified in the spell (4d8 lightning) "instead of the weapon’s normal damage." The weapon's normal damage may include your Dex (or Str) modifier and may include sharpshooter but the spell damage doesn't.