Even after cancelling the EntityDamageByEntityEvent, entities are still inflicted fire ticks when damaged by arrows enchanted with flame. Tested on a server with no other plugins but this one. I expect the flame effect to not apply because spectral arrows and tipped arrows do not apply their effects when cancelled. All code in the plugin is included below as a simple test case:

public void onEnable() {
    Bukkit.getPluginManager().registerEvents(this, this);
}    

@EventHandler
public void onDamage(EntityDamageByEntityEvent e) {
    e.setCancelled(true);
}

See attached video for example.