I think if high-end analog oscilloscopes had continued to be developed into the late 90s, screensavers for the CRT might have become a thing, but for just a plain old analog oscilloscope it seems like a lot of work to implement. The only ones I can think of (as already mentioned) are the Teks with MCP.
MCP oscilloscopes are a special case because the MCP has a limited operating life.
As said above, just blanking the CRT display would have been an easy way to avoid screen burns.
You have to be careful about blanking a CRT. Extended operating time with the heater active but no cathode current can damage the cathode. (1) This becomes a problem in applications where the instrument is powered for long duration but the CRT intensity is turned off to preserve the CRT. Turn the intensity down, but not off, and defocus it if you must.
Vacuum tubes have the same problem when operated in cutoff. Special cathode materials were used to avoid this issue in applications where operation in cutoff would be routine like digital logic.
The above suggests that analog oscilloscope CRTs were constructed to operate in cutoff, at least after the problem was recognized, but *raster* CRTs were not.
(1) Robert B. Tomer calls this "interface resistance" in his book but does not describe the mechanism because I do not think anybody at the time knew how it happened. It results in a severe loss of transconductance. My guess is that ion poisoning is responsible when the ion trap is not operating.
The so-called "ion trap" used in earlier TV CRTs doesn't trap anything!
Before such devices, CRTs suffered from "ion burn", where a small circular burn appeared at the centre of the tube over time---- it looked uncannily like a cigarette burn!
Along with the desired electrons, cathodes emit
negative ions, which are also attracted to the screen due to its high positive potential.
They are not deflected by the magnetic deflection systems used in TV tubes, so always impact the phosphor in the same spot.
Although they travel slower than electrons, they are much more massive, & over time, damage the phosphor.
(In the case of ordinary vacuum tubes, the anode is mechanically rugged, & no damage is done.)
To solve this problem, CRT manufacturers positioned electron guns so that they were "pointed" off screen.
To get the electron beam centred, they used a fixed magnet to deflect it back "in line".
The negative ions were not so deflected, & continued on their way, finally harmlessly impacting upon the inside of the glass envelope.
There remained one other problem---- when the TV was turned off, the desired electron beam was now undeflected, the cathode was still hot, & EHT was still present, resulting in an intense electron beam hitting the centre of the tube.
This only happened at turn off, & took a lot longer to show ill effects than ion burn, but would show eventually.
This was prevented by the use of a "spot swallower" which opened the cathode connection at turn off.
All this technology cost money, adding to the final price of TV set, but aluminised screens swept away both problems "in one fell swoop".
The only mechanism by which ions could damage the cathode, as far as I can see, would be if the grid was biased well negative of the cathode, where the ions, once emitted, would return to the cathode which is now the more positive tube element.
Positive ions are not present in any significant numbers in the high vacuum of a CRT.
Leaving a CRT or any tube on heaters/filaments only on for a long time while not in use, expends lifetime "hours" as the emitting material has a limted service life before emission starts to fall.
That said, older Radio & TV Broadcasting transmitters using all tube construction commonly had standard "warm up" times of around 10-15 minutes.
This was mainly due to the need to fully prepare mercury vapor rectifiers for use.
All the other tubes shared this time interval, with no effect upon their life times.
OK, that is not quite the condition you referred to, which would be with all the other elements powered as normal, but the grid biased to cut the tube off.
TV sync generators happily operated multiple bistables using 12AT7s or similar, with, at any time, one triode of each pair in cutoff.
They also were used in monostables where "one triode in cutoff" would be for more extended periods.
This would be similar to your suggested "data" use.
Again not quite the use case you mentioned.
On old analog scopes which basically had no "fixed" area of the display, the problem would be minor unless the user left the exact same signal on display for a very long time. Probably not that much of a problem in practice.
The problem started to appear when scopes started to have on-screen display of extra information. I guess automatically blanking the screen after a defined amount of time would have solved the issue, but OTOH that could have been pretty annoying for many use cases.
The common aluminized phosphors used in oscilloscope CRTs are not susceptible to screen burn except at very high beam currents which explains why I have never seen ones of these CRTs with screen burn from the readout. Contrast them with storage and MCP CRTs where screen burn in the readout area is common.