The reason that it's not a good idea to run a car's tank dry is that the catalytic converter can be damaged. Whether or not that is the case I don't know but that is the common advice from manufacturers in the handbooks of all cars I've either owned or driven.
Aircraft of course don't have cats so that removes that problem. As for deliberately running a tank dry, personally I wouldn't do it unless I was in major bother and trying to squeeze every last drop out of the tanks to get home, shouldn't be a problem in ye olde carburetted Yank flat fours most of us sit behind but I'd still be worried about the engine not restarting...
Smithy