Are Final Tests always required for LRUs?
I'm assuming by Pre-Testing procedure you are referring to a physical inspection and downloading the fault history.  

If there are no logged faults in the fault history, it means that the built-in test (BIT) of the LRU did not detect any faults. BIT is not capable of 100% test coverage. It is also possible that BIT results are inaccurate. A lack of recorded faults is not sufficient to declare a unit airworthy. 

Once an LRU is tagged and sent to the service center/shop, it can only be returned to service (RTS) after it has passed the RTS tests specified in the component maintenance manual. After passing the RTS tests, a 'maintenance release document' can be prepared and the unit returned to the spares pool.   

In the US, the maintenance release document may be an FAA Form 8130-3, Form 337, or other document containing the information required by 14 CFR Part 43, sections 43.9 and 43.11. The maintenance release document must be signed by an authorized person who is certificated under 14 CFR Part 65. Though not legally required many shops will affix a "yellow tag" to the unit to indicate it's serviceable status.  

EASA has similar rules, the primary difference being the required content of the documentation.