It depends on how the ink cartridge/print head is designed. Some ink cartridges have the print head built in. Since you typically throw them away when they are empty, running them dry and damaging them isn't really much of a concern.
If the print head assembly is separate from the ink tank, running them dry can result in clogged jets or transfer pads. They might be recoverable if replacement ink is added to dissolve the dried ink. Or it can result in more expensive replacement parts.