For completeness's sake,
Fire Shield
"...In addition, whenever a creature within 5 feet of you hits you with a melee attack, the shield erupts with flame. The attacker takes 2d8 fire damage from a warm shield, or 2d8 cold damage from a cold shield."
Spiritual Weapn
"...You create a floating, spectral weapon within range that lasts for the duration or until you cast this spell again. When you cast the spell, you can make a melee spell attack against a creature within 5 feet of the weapon. On a hit, the target takes force damage equal to 1d8 + your spellcasting ability modifier. As a bonus action on your turn, you can move the weapon up to 20 feet and repeat the attack against a creature within 5 feet of it..."
Case: Cleric > 5ft from enemy
If the Cleric was greater than 5 ft away then the answer both RAI and RAW is no. The creature attacking must be within 5 ft. Clearly the Cleric is not within 5 ft and thus takes no damage.
Case: Cleric = 5ft from enemy
I believe the line I bolded for Spiritual Weapon above is simply a game rule on how to resolve the hit/miss of the spiritual weapon. The spiritual weapon is still the 'thing' doing the hitting and not the cleric. Thus, I believe RAW supports the ruling that even when the Cleric is 5ft away and uses his bonus action to cause the spiritual weapon to attack (*and it hits) that Fire Shield would not do anything because the Spiritual Weapon is not a creature and Fire Shield requires a creature within 5 feet of you to hit you with a melee attack. I also believe ruling the other way is not against RAW but does cause fictional issues - which makes in an inferior ruling IMO.