I only ever heard of running out of fuel cause damage once.
A guy I work with had an old Rover 214, he ran out fuel and then flooded the engine by pumping the accelerator too much when he re-fueled. He let it sit for 10 mins then it started OK and all seemed well.
When he went for his next MOT about six months later it failed the emmisions test. The MOT station said it was probably down to raw fuel destroying the material in the catalyst, some fuel had entered the exhaust and must have reached the catalyst, which apparently is a bad thing.
With hindsight he said he had developed a rattle in the exhaust which he assumed was a heat shield rattling, but it turned out it was the material in the cat which had disintegrated. The only time he could think of that fuel might have got into the exhaust was when he ran out of fuel and then flooded it. A new cat was about £250 at the time so it was an expensive MOT for him.
Not sure how likely this or if the MOT station was taking the mickey, but he maintains that he will never run out of fuel again which with the other mention of crud in the tank might be the best policy overall.