If they can survive long enough to allow the fuse / MCB to cut the power then the chances are the item they are protecting will not be harmed.
..but the chances are the MOV will be taken out, not able to provide any further protection and the operator will not be aware of that fact. In Gasguru's example, the sequence might well have been -
1 MOV provides protection once against the first surge and self destructs.
2 A second surge then takes out the PCB, because it no longer has any surge protection.