Hi Danny,
the multiplexer function switches X-input signals to one output. To test the function of a digital multiplexer each channel needs to be selected an toggled with a Low and High logic level and the changes can be test on the output. To detect also shortages between the channels the other inputs should have the opposite logic level. If you need to cover all failure modes you have to do an FMEA analysis at the iC-level. Typical signs of a damaged chip are high static supply current or high input leakage(e.g. ESD damage).
Do you want to test an analog or digital multiplexer?
Enjoy your design work!