Zener voltage regulation happens in reverse breakdown mode. All zeners leak reverse current at voltages below the breakdown voltage. With low voltage zeners, this leakage current is very high, up to 76mA for this part number. This makes low voltage zeners unsuitable for many applications, where you might expect a theoretically perfect zener to work.
If the 50mA or so leakage at 3V is an unacceptable issue for you, you might instead consider a potential divider across the 3V supply, with the centerpoint connected to an SCR gate to crowbar the supply. Adjust the potential divider ratio to give the right tripping voltage. A red led (forward biased) added at the top of the potential divider might make it more accurate/sensitive.
I haven't actually tried this at such a low voltage, but I suspect it may work better than a zener.