As a home inspector in San Diego I always do, and why wouldn't you? This is part of your trade and probably is in your standards of practice if you belong to one of the associations like NACHI or ASHI. It is very simple to test and it either works, or it doesn't. There is no need for further testing to determine why it does not work. You would simply recommend further evaluation by a licensed electrician and leave it at that.
The electrician will probably follow a fact finding mission such as below to determine why the bell is not ringing.
There are a few components of a door bell system that can be inspected. First, you can remove the button to be sure the wires are attached and not corroded which happens frequently. Next, you can attach a signal device to the wiring and test to see if the signal is picked up at the transformer. This tests the continuity of the wiring. Next, test the transformer by testing the input and output voltage to see if it is working properly. Finally, remove the chime device (the bell) and check to see that the wires are attached. Have someone push the button and see if electrical current travels all the way to the chime. If you are good up to this point and the bell does not sound, the chime itself is probably faulty.