IMO, running with tyres of slightly different diameter won't generally do your vehicle much harm, though you wouldn't want to do it for thousands of miles. It's worse if the tyres differ left to right, than front to back though, so a single spare of an odd size isn't ideal. This is more an issue of unevenness in braking and handling, than differential problems.
If the tyres differ in diameter by, say 5% (i.e. a 31.5 inch and a 33 inch tyre), then the diff pinions will need to rotate at 2.5% of the road-wheel speed (one forwards, one backwards). At 120km/h, the road-wheels are rotating at around 800 rpm, so the diff pinions "spin" at around 20 rpm - or one revolution every three seconds. Nothing's going to get hot or wear out very fast.
Even with permanent 4WD, the centre diff isn't working very hard - a little faster than one rev per second (or half that if there's only one odd-size wheel).
Having said all that, there doesn't seem to be all that much to gain by carrying an odd-size spare wheel on an expedition vehicle. The original context for my thinking about this issue was whther it is essential to use a tyre size that is readily available in the countries you are visiting. I decided I could always get away with using a locally available 7.50-16 as an emergency replacement for the rare 255/85-16s.
Naturally, locking the CDL or engaging 4x4 on a part-time system should only be done when on a slippery surface (and axle locks even more so) - the 5% difference equates to making 40m sweeping turns, so it's not a good idea to lock anything unless the ground will yield easily.
Btw, I have no idea how a viscous coupling would enjoy a constant speed difference of 5%. Any ideas?