Is an Arduino Uno capable of running 24/7?
I have powered the board for days at a time.  The code that was running was very simple, but there was absolutely no damage.  It is worth it to note that it was being powered by a pre-regulated 5v source so the on-board regulators were not burning up. 

I doubt that with anything lower than 9v there could be any sort of hardware damage, but with larger voltages the on board regulators might start to get very hot.