Hi,
How are you powering the relay?
If you have a diode across the coil, you can ONLY CONNECT IT ONE WAY .
Positive to the coil terminal with the Cathode of the diode connected, and Negative to the terminal with the Anode of the diode connected.
If you connect it the other way it will not operate, and will appear as if you have a short circuit coil.
If your power supply is able to supply many amps, then you will destroy the diode and it will appear as a short circuit in either direction.
Tom... 
Does your DMM have a diode testing function?
Can you tell us the model of the DMM, or a picture?