Well you could argue about this for any other piece of test equipment, or any other kind of electronics.
Turning things on and off does put electronics trough extra stress as it might transition trough some rough events during it and the effect of thermal expansion as it heats up and cools down can put stress on various parts of it. The thermal cycling can also cause references such as crystal oscillators of precision voltage references to drift more than usual. Some equipment also needs a lengthy few minute warmup before it provides the most accurate of measurements. So there are reasons why you might keep a piece of test gear running 24/7
But then again running constantly racks up extra operational hours. This could cause more deterioration in some components such as capacitors, fan bearings etc. Displays also tend to have a lifetime, be it the segments of a VFD display, the phosphor of a CRT screen, the CCFL backlight tube, or the LEDs in a LCD backlight. So plenty of arguments of why you might not want to run something 24/7
Hard to say what is better for this particular network analyzer, but don't worry about it and just do as you are told. If it breaks its going to cost the company to repair anyway. Tho if the reason it broke is because you sent 100W of RF into one of the test ports, then maybe you should worry at least a bit as these thing was surely not cheap.