it could if the other tires are worn down, but it would be unlikely. a better question would be why rotate the spar into the mix unless you like buying 5 tires at once, dosnt really serve much purpose unless your only replacing 1 tire and working that in to save the cost on a tire and making an old one the spare
Unfortunately, it seems that a high percentage of mechanics don't know which end of the wrench to hold, and the service writers absolutely don't understand the first thing about vehicles and how they work. This is true across all brands, apparently. Sometimes there is an older mech around that knows what he is doing, but usually not.
I do five tire rotation on my LC. Sometimes I do it and sometimes the tire shop does it (free there since I bought the tires from them).
5 tire rotations are perfectly fine so long as they're the same tire size. If you're this concerned with wear amounts and their impact on differentials, then you should be fastidious with checking your tire pressures as they vary more frequently and are more likely to cause more height variation.
Unless you're running a true locker (i.e. Detroit, engaged e-locker, etc) your differential will not see harm by these slight variations. It's literally built to account for this.
Meant to say this earlier.. I'm pretty sure the dealer was on some level knowingly lying to you about the potential damage, so they could be lazy and not have to drop the spare.
Back to what was said above about never listening to the dealer.