The mute actually shorts the leads of the cartridge so the test is to use a simple VOM to measure the resistance of each channel which should be ~1000 ohms when sitting on the record. 

So measure between pins 3 and 2 (gnd) for Left channel and 5 and 2 for Right and should be ~1K Ohms?  That's a DIN connector pictured here but the unit has RCA plugs so measure between center pin and ground at the output?

Yes, between pin and shield of each RCA. 

Does it make a difference if the cartridge is installed when I test?  I assume so...

Yes it needs to be installed and lowered to play a record. 

Connected the Simpson Meter and measured 0 Ohms before the record is played.  As soon as the record plays the resistance goes to infinity... happens on both channels at any R setting (x1, x100, x10,000).  I can hear the relay apparently operating at the correct time.  If I'm looking for 1K ohms something isn't right. 

I tried both mmc4 and mmc2 which both produce sound properly on my Beogram 3000.

The fact that both channels are open (infinite resistance) is point to your tonearm wiring. That wiring is most vulnerable in the area of the tonearm bearing assuming no one has been inside. I’m afraid you will need to physically inspect along its path and will likely need to repair as I have only ever retired their linear tracking models which is easier. This is job for a qualified tech as trying to rewire the entire arm without damage requires removing the glued cartridge mount. Given the age of the table if the break is not in an area that can be spliced, I would recommend finding a replacement tonearm from a parts unit. Once again this requires an experienced tech.