Could Removing A Nintendo Game Boy Game While The Console Is Powered-On Do Any Damage?
What's up?
No, not beyond save data corruption.
In fact, the Game Boy's big brother console, the Nintendo 64 had games that planned on using live cart swapping as a feature, to link games (most famously with Banjo Kazooie/Banjo Tooie), but it was scrapped due to other concerns (essentially ram being too volatile for it to work reliably).
There is one caveat: Poorly designed cartridges could be knocked out of commission by corrupted save data alone.
Note also: All of this applies to the OG Game Boy, the Color, the Pocket, the Advance and consoles NES, SNES and N64. From DS onwards, the system is designed specifically to detect cartridge removal and handle it gracefully; and the GC/Wii (U) don't use cartridges anymore.
So... why?
The question is, of course, why do these systems tolerate this at all? The basic reason is that in straightforward terms: There's nothing there to damage. The consoles mentioned above are, in essence, just a circuit board with a processor on it, some ROM chips that provide libraries, some volatile RAM, I/O, and a power supply. There's no OS or persistent storage or really anything else. When a cartridge is plugged in, it's plugged into what essentially works as an expansion slot on this board, providing the bulk of the actual executable code in ROM, as well as anything else the cartridge might need (such as flash or battery RAM memory for save files).
Most of the systems have physical barriers to removing the cartridge (usually the power button engages a lever that inserts itself into a groove in the cartridge, preventing it from being removed (or inserted) while the power is on).
The first console Nintendo shipped without a physical barrier like this was the Game Boy Color, which allows a cartridge to be hot swapped; as far as I recall it detects this situation and gracefully shuts down when this happens. This is likely a feature that wasn't present in the earliest Game Boys or home consoles due to the physical barriers simply being easier to implement; but the proliferation of different-shaped Game Boy cartridges made the existing safeguards ineffectual or counter-productive, and the console was made to detect the situation and handle it instead.
In any case, the simplicity of design means that there's not really anything for pulling out a cartridge to damage in any meaningful way; which is a premium design feature in a product designed to be roughed around by kids.