Here on LG, as well as on GoL I've seen many cries about protected programmed start electronic ballasts with some partially worn, but still good lamps sometimes only flash after preheat at power ON, but do not continue to light the lamp.
This weekend i messed up with some dimmable ballast - converting it to "diode coded" 3-level dimming system (uses original single circuit wiring, but by the polarity 3 light levels are "coded", e.g.: positive only = lowest, negative only = medium, both polarities = full power; in the ballast is then incorporated a "decoder" and this was the the part i was adding).
When testing the final ballast assembly (with the added "decoder"), the EOL protection start to trigger, but only when started at Full power and when the ballast was still warm. When starting at Medium (~27%) or Low (~8%) setting and few seconds after it light up switching to full power didn't trigger the protection anymore.
I find out, then the root cause was wrong capacitor type (some Z5U or Y5V instead of X7R or X5R) in the preheat timer, what had lower capacitance at higher temperatures.
Consequence was, then the Preheating phase was too short, so filaments didn't reach required emission temperature, when the controller enter the Run mode and enabled EOL protections.
In my case the emission was sufficient for reduced power setting, so at Medium or Low it successfully started, but the emission was not sufficient for the high current of the Full setting, so the protection detect it as EOL. After few seconds even at Low or Medium setting electrodes reach the correct temperature, so switching to Full power caused no issues anymore.

So ballasts, that trigger the EOL protection on still good, only partly worn lamps, likely have insufficient time for Preheat phase - or by design error (was designed as marginal and with some component tolerance and lamp wear it become insufficient), wrong component type used (someone tries to make it cheaper in production and does not check all relations) or a defect.
In all cases it is quite simple to fix:
Identify the Preheat timing capacitor (consult controller chip datasheet)
Replace it with some of ~1.5..2x higher value (and be careful to use only thermally stable type - film or X7R ceramic, better do not use electrolytic)

Prolonging the preheat time does not hurt (in contrast, insufficient time is very damaging; The preheat current value is more important, so would not touch this), it is only matter of slight (and only hypothetical) discomfort in waiting a second more for the lamp to start (here i prefer long life over one to three second delay).

Is there any way to "disable" or "override" the EOL protection?  I would hate to deal with such a nuisance, especially if the whole ballast shuts down if one lamp goes EOL.

@DieselNut: Only those requiring external (from the controller IC point of view) components, so only lamp voltage sensing (very frequently use resistive divider and pair of antiseries Zener diodes tied to Enable/EOL input of the controller). But you should make sure you do not disable the feedback loop for Preheat and Ignition voltage limitation (if i remember well, some Philips/NXP chip feature this) if implemented (some chips/designs feature this to allow more reliable preheating and component overstress).

But beside the (rather optional, my ballast don't have it) voltage sensing, virtually all controllers feature some form of Hard Switching detection (usually somehow linked to Overcurrent protection) and this is at first impossible (mostly) to disable without affecting normal ballast functionality (e.g. IR21592) or hardwired inside the controller chip (e.g. IRS2530).
And as second, I would not do this anyway, because hard switching is, what destroy the inverter output transistors.

But i do not know, what do you want to achieve: Or really run lamp, till they loose vacuum (strongly discourage this except test bench under supervision - tend to melt tubes out of sockets), or deal with the problem of false tripping EOL protection?

For the first better use some cheap self oscillating ballast.
And for the second the easiest would be (as i see it) to try to prolong the preheat time and as second easiest is disabling the protection.
If one inverter feed two lamps, you have to replace the dead lamp anyway. What might help, replace always both as "group" with new lamps: Both have the same hours, same amount of start cycles, same operating conditions, so if one fail, the second has only small fraction of it's life left, so it would cost practically nothing (both $$ as well as environment impact) to replace it as well.
If you really need independent lamp operation, you have to use one ballast per one lamp.