I do not have a DRO, but have worked with electronic equipment all my career and much of it was run for long hours each day, if not 24-7-365. Some things were never turned off unless there was a power failure and others were off at night and on again in the morning. Since it was my job to maintain the equipment, I did observe the failure modes.
With a DRO, which is essentially a digital device, I would be concerned with two possible long term failure modes. The first would be power spikes on the incoming power lines. When there is an electrical storm, lightning bolts can induce high Voltage spikes in the power lines and those can get past the filtering in the power supplies in the equipment. Digital devices work on Voltages of 5 Volts or less and many of the chips can fail if the Voltage exceeds 8 or 10 Volts. Lightning bolts exceed 100,000 Volts and if even a small fraction of that gets to the chips, they are toast. I have seen it happen. Turning the power off can help to prevent this. Analog circuits are more resistant to this kind of problem.
On the other hand, parts like the electrolytic filter capacitors in those power supplies will age. One failure mode is a sudden short when power is turned on. I have seen electronic equipment that is always left on fail when it was turned back on after a power outage or it was turned off for some specific reason. This aging is in proportion to the hours of use so any hours that the equipment is off will lengthen it's life.
One other factor that should be considered for some equipment is thermal cycling. If you turn it on and off, it will heat up and cool down. This can produce additional stress for some components.
These are not DRO specific factors. They apply to electronic equipment in general and often to digital equipment more specifically. It is an open question as to which is the better strategy. I have seen electronic and computer professionals who advised everyone to turn their equipment off every night and others who advised that they be left on because "most failures occur when it was turned on". Personally I have always turned things off when they were not in use. This has around 50 years of experience behind it but no real scientific data that I am aware of.
A side story: the old tube type equipment had higher Voltages (125 to 500 Volts or even more). It also had power supplies with electrolytic capacitors and they also aged and failed. But the failures could be much more spectacular with the higher energies that were available in them. I once had an electrolytic capacitor in a 450 Volt tube type power supply erupt like a volcano ... MY FACE WAS INCHES AWAY. I would have had severe eye damage, possibly even been blinded if it were not for a steel rail that was in front of my eyes. The debris hit me above and below them. I was quite shook up. It was probably around 20 years old and this happened when it was turned on after a period of storage.
