What causes a varistor to have low resistance when connected to a board?
The board is causing the difference in the measurements.  More precisely, other components on the board that are connected to the MOV.

The MOV only reacts to high voltage.  Since the board is powered off when you are measuring things, it can't be the MOV itself.

If you don't have a circuit diagram (or can't trace the circuit yourself) then you can't tell if that low resistance reading means something is broken.

Since the MOV reads as an open circuit off the board, it is probably OK.

I can't tell you what might be wrong with the board. That would take more details than you've given - and maybe more details than is practical over the internet.  At a minimum you'd need a diagram of the circuit board - either from the manufacturer or traced by hand from the board.