Do I need to remove the varistor from the circuit to check its validity with DMM?
The varistor (INR 14D681S) reads 0.650 MOhms while it is attached to
  the PCB.


That seems about ball-park correct



The data says that the DC voltage that produces 1 mA current is 680 volts (as per the "681" in the part number. 680/1 mA = 0.68 Mohms.

Your meter read 0.65 Mohms so it's about right. If you pulled a leg out it might read a few Mohms BUT, the important thing here is that you haven't found a problem with the MOV with your test.


  Do I need to detach it &amp; test it with DMM if there's no visible sign
  of damage on it?


If it's failed then your multimeter test will not show this whether it's in-circuit or sat on your bench. At the moment, it looks OK but, importantly what are the symptoms that caused you to suspect this component?