What is the cause of the error/unexpected outcome in water temperature control heating system circuit?
In 50.6˚C, VA=4.25V, VB=4.58V, ONLY green lights up (unexpected outcome). 


There is enough current going through the relay to light up the green LED but not enough to switch the contacts of the relay, I suspect the current going through the LED is between 100 uA and 10 mA. That is due to the output of the opamp, which is probably activating the transistor in its active region.

One more thing, what do you mean by this


  the diagram,the orientation of the green LED is wrong, the positive terminal should connect to relay.


The Green LED is fine as shown in the schematic.