Continuity tests, risks?
Continuity testing of an in-circuit component is not a reliable procedure, independent of signal injection and its associated risks. 

The component leads may be interconnected via other circuit elements, thus giving a false continuity result, where the component itself is actually not a conductive route.

Regarding the risks of introducing a voltage through the multimeter leads:


Yes, there is a risk of certain parts being damaged, especially parts that can not tolerate the 1 to 9 volts that a multimeter might deliver across the probes in continuity mode. 
The above is especially true when the component (or other components on connected traces, which will also be affected) is not powered. Many parts can tolerate voltages when powered but not otherwise. 
To minimize the voltage, an option is to use the multimeter in resistance mode, at the lowest resistance setting - The higher resistance scales work on higher probe voltage, going by a quick check on a couple of multimeters at my desk.
Note that basic multimeters often combine continuity and diode testing modes, so the voltage is at minimum sufficient to forward bias silicon diodes and perhaps LEDs. This means a voltage of 2 to 3 volts.


In summary: This is best not done unless the experimenter is open to the risk of damaging the device in question - not just the component being tested, but other parts of the board.