Something I've been wondering about for a while is what would happen in a fixture where one ballast drives two lamps, if one of these lamps was spot replaced for an LED tube while the other retained a fluorescent tube?

It depends heavily on the ballast schematics, and if it is magnetic or electronic. Generally, it has better chance to work OK in parallel circuits.

I was just staying in a hotel room that had that very setup.  The kitchen area had a troffer containing what I assume to be a regular magnetic rapid start ballast, which has both tubes operating in series.  One tube was a regular F40T12 fluorescent tube and the other was a Philips "Universal Fit" T8 LED tube.  Seemed to work just fine, with the fluorescent tube having typical rapid start startup behavior and the LED tube coming up to brightness with it.  If I unscrewed either tube, both would go out.

From my experience, this is becoming more common in buildings that do not have regular re-lamping. When spot replacing lamps, some places get LED tubes instead of fluorescent. This seems to work most times.

And I also think it will vary among different LED products, what they are designed for. It is technically possible to design the PED ballast to work on any fluorescent ballast you pick, but youhave to chose which one, it is not possible to have it compatible with all of them (the original fluorescent tube rating is not enough). Generally only few are commercially made, so you need to check with the LED product datasheet, whether it is compatible with your fluorescent ballast.