Is it safe to run Arduino indefinitely? Does it ever need to be switched off?
Afanasjevas:
Is it safe to run Arduino indefinitely? Does it ever need to be switched off?
Yes, it can be run continuously with no problems. Indefinitely is not likely, as our Sun will be exploding in the future and taking the earth with it. 
Lefty
Most of my projects are intended to run continuously, I don't seem to buy a lot of power switches these days!
If you use the millis() or micros() functions you will need to consider that they rollover and start again at 0 periodically. The millis() function is good for about 50 days before a rollover, the micros() function about 70 minutes.
kuhntucker:
If you use the millis() or micros() functions you will need to consider that they rollover and start again at 0 periodically. The millis() function is good for about 50 days before a rollover, the micros() function about 70 minutes.
The roll over will be no problem if you follow the techniques in the Blink without delay example.
our Sun will be exploding in the future
I think exploding is a bit too dramatic, it is more of a slow expansion that will see the Earth being swallowed up. It happens over many Melania so it won't be sudden, still a bit catastrophic mind you.
The great Zeusarus has already diminished taking it solar system engulfed by expansion and then burst out all the grey matter spreading and moving toward the Earth at the speed of photon with infinite mass... Seems the sun will have no life for decay and expansion.. 