I'm sure tyres can survive a small drop off a ledge if they can withstand landing forces. The main concern would be fore-aft strains on nose gear (strut assemblies, etc) if being towed or fore-aft strains on the main gear if being powered out.
Three points here.
Tyres. Quite!
Nose gear. The towbar fuse prevents this. Not a factor.
Main gear. Thank God landing gear isn't that weak or we'd have a collapse every time we landed or put on power against the brakes. Surely no one is suggesting that gear that repeatedly withstands the phenomenal forces of landing and emergency braking including landing with drift etc could possibly be damaged by power against brakes? How could full power engine runs ever occur if this was even remotely possible? (and before some expert pops up with only "but that's only one engine at a time" that results in the same forces on the main gear leg on that side - on a twin...). Anyway, no one has even remotely suggested using TOGA power to break out of the mud so this fanciful scenario is completely irrelevant.
Aircraft occasionally jump chocks - does that damage the gear? It may damage other things, wingtips, vehicles, laundry etc but not the gear.) And as this is achieved relatively easily under power why would/could motoring out of a similar sized (depth) depression in earth back onto the concrete damage anything?