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.

A couple of things to keep in mind (outside of @Sachleen's mention of millis()):

• Like any electronics, heat can be disruptive. The micro-controller itself isn't likely going to be a huge issue from the perspective of heat, but other components like the power supply might cause issues.

• If your code uses EEPROM.write(), be aware that the EEPROM in your Uno's ATmega328P is only rated for 100,000 writes.

Keep in mind that the flash and EEPROM have limited lifetimes (about 10,000 and 100,000 write cycles respectively) so if you're doing a lot of writing to those, they may become corrupted. In a test I did, an external EEPROM took about 3 days to start becoming corrupted.