Arrows are projectiles that impart kinetic energy into their targets. Said energy causes damage to their structure. Metal is perforated, flesh is cut, bone is broken. No matter what the target is made of, it is still suffers physical damage. ...
So no, I have no problem with shooting robots with arrows.
I do. Its not always true that a thing always takes damage. It depends on the thing and what's 'damaging' it. You can hit a tank for a week with a baseball bat and destroy the bat with no damage to the tank. With some materials there is a threshold where any damage below a certain force is always cosmetic or nothing at all. Now if the robot is made of Balsa wood, or tin, that's different, but if it's a strong metal ~even magical craft? It would surely take more than a stone tipped arrow to bother it. Even a steel tipped spike on the arrow might have trouble punching through ½ inch metal plate. (Of course, we can't know if they are ½ inch plate, but they don't look like they are foil covered.

)
Hmm, let's think about that. An ordinary solid, metal object vs magic... Maybe because a fireball is magic?
Is it magic?
(Or more specifically... is it magical fire, or normal/regular fire brought about by magical means?)
In my thinking... If a mage cast a spell that opened a 'hole' into a fiery dimension, and out shot a super heated stream of conventional flames ~it still should not effect a ghost.

* But an enchantment (of magical fire?) intended to harm a ghost could 'burn' one ~and perhaps set fire to nothing else in the area; http://i271.photobucket.com/albums/jj125/Gizmojunk/magic.jpg.