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It is conventional wisdom for a reason. When I worked for a transaxle manufacturer we did accelerated differential testing by running different size tires.
How much damage you do depends on the difference in tire sizes and the design of the drivetrain. A simple open differential will take abuse much better than a differential with clutch packs like a LSD. The new fancy differentials that can selectively send power to individual wheels using clutches - you really don't want to burn up those clutch packs.
With an AWD car you are stressing two differentials by running an underside tire.
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