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.

I've had one operating a simple weather station which sat online for a couple of months without any problems - I don't see anything that would cause it to break down beyond that time either.

The only real killer (aside from external factors) would be heat, so I'd advise trialling it for your usage application and seeing how it does. If you're worried it's getting too hot, then adding a heatsink shouldn't be too hard an addition to stop temps becoming a concern.

Keep in mind that the Arduino is meant to be used as a prototyping device. This means there has been very limited endurance testing of the board.

Once the board reaches a steady state temperature, there is nothing in the design that would cause it to go into thermal runaway on its own.

How you program it and what you hook it up to, however, may be a different story.