Can a DMM continuity probe fry an IC?
DMM diode test or continuity currents are typically 1mA which is well under the 5mA guaranteed level for ESD protection diodes.  Latchup only occurs if power is on and any input or output exceeds the supply rail by 0.7V

However if you failed to discharge your meter and yourself to ground before probing, you may have exceeded the 3kV ESD protection for a std 100 pF rating, for a standard finger.    

No Risk, touch gnd pad first. or use strap.