Does using a straight through cable between two similar network devices (switch to switch, router to router, PC to PC, etc) cause any damage to the switch or the router that you are connecting? I understand that you must use a crossover cable to connect two similar network devices together.
No, it wouldn't cause any damage.
But all modern Ethernet cards support automatic crossover (Auto-MDIX), which means you can (and in many cases, should) simply use a straight-through cable and the two adapters will automatically decide how to use each pair. This feature is present in practically all Gigabit Ethernet adapters as well as huge majority of 100 Mbps "Fast Ethernet" adapters.
(Gigabit Ethernet actually uses all pairs in both directions at the same time, so if your idea of a 'crossover cable' is to just swap 2 pairs, that no longer makes sense anyway.)
What can cause damage is connecting a cable from a proprietary "Passive PoE" adapter to a device that isn't expecting PoE.