Zener diodes in glass axial package - not inherently shielded from photoelectric effect?
Diodes of all sorts, including the ubiquitous 1N4148, packaged in transparent packages tend to have some sensitivity to light (both photoconductive and photovoltaic as you have observed). The 1N4148 can apparently produce 10nA in direct sunlight. 

I rather suspect your zener diode when used normally with several mA flowing would have negligible response to normal room light. Zeners are not terribly precise devices in the first place. However, say you are using it as a noise source, say for audio or cryptography, you might want to keep it dark or use a plastic packaged device. 

It's worth considering such effects if you have a very sensitive circuit and it is exposed to light, either from openings in the enclosure or because some designer has peppered the PCB with highly luminous LEDs that are modulated or blink. 

That includes glass MELF packages as well as axial-lead packages (photo from Digikey).