So the neck of my Zenith glows and I know that's to aid in bring up a picture as soon after powering on as possible. However, I was wondering if this element has a lifespan and what happens if it stops working? My set is 51 years old and sometimes it sits for months without being turned on ( I know, that's abuse). So, should I unplug it if not using for extended periods?

Yeah turn off that instant-on feature or unplug it, not only do they have a finite lifespan but I've had cathodes "poison" over time leaving them warm without any current on them.

So the neck of my Zenith glows and I know that's to aid in bring up a picture as soon after powering on as possible.

If we're assuming that you mean the neck glows when the set it plugged in but not turned on, then I will echo what ARC says.

It can eventually burn out, but it's a lot more robust than a light bulb filament, and it's run at a much lower voltage.

The real danger is that the heater is spending the barium from the cathodes. As the barium emits its electrons, the outer surface of the cathodes develop an inactive area that doesn't emit anymore. When that happens, the "normal" amount of screen voltage will no longer cause the same level of cathode emissions and the image appears dimmer.

Instant on can be good for the tube when the set is on one or more times a day but it's bad for prolonged sitting out of use.

Instant on can be good for the tube when the set is on one or more times a day but it's bad for prolonged sitting out of use.

Has there been a study done on the effects of heat cycling VS up-time on cathode material?

well that settles it, off it goes.