if you short between pins 30 and 31 of the 7106 (IC3), and the displayed reading on the display drops to 0.00 (or close thereto) then the fault almost certainly lies in the switching circuitry.
attached is a marked-up copy of the schematic for when the 2v DC range is selected. RED is the positive input path, passing through the 9M dropped resistor (the upper half of a 10:1 divider), and through various switches to R23 and then into pin 31 (IN+) of the 7106.
BLUE is the ground path, going to pin 30 (IN-). pin 32 (COM) is also tied to ground, which is an OUTPUT from the 7106 that it holds a (precicely regulated) couple of volts or so below the +ve battery terminal. ground also connects via a string of resistors adding up to 1M that runs up to the bottom end of the 9M resistor. the string of resistors RP1, R3, R4, R5, R6 form the input voltage divider.
YELLOW is the 200mV reference input to the 7106.
the only other thing that could possibly cause problems is IC4. it controls the values of integration components attached to pins 28 and 29 of the 7106, essentially changing between 200mV full scale and 2V full scale. this is likely something to do with the ohms ranges, and even if not switched correctly would not affect zero on volts or amps ranges.
(addendum) however, if IC4 were defective and injecting odd voltages at its pins 1,2,3,4,5,13, then this could cause non-zero display with no input to the 7106. but i'd consider it an unlikely fault.
cheers,
rob :-)