Is an Arduino capable of running 24/7?
You shouldn't have any issues keeping it on all the time, however, some thing to consider is any counters you may have, like the use of millis().

From the Arduino docs on millis:


  This number will overflow (go back to zero), after approximately 50 days.


So for projects that are on for long periods of time, you may not see an issue immediately but something like this could pop up and cause errors down the road.