I don't think that the low current of the multimeter can destroy it. But the datasheet of the dsPIC30f6010A says: "The analog inputs have diodes to VDD and VSS as ESD protection.". Does this mean, that the digital inputs don't have ESD protection? Then this could be the reason, when touching a contact with a finger.
I've not used dsPICs so far, but the PIC 16F and 18F series and never had such a problem, even when I bridged incidentally a GPIO to VDD or VSS.