Does a delay in a loop save energy? I've noticed most example snippets always have a delay in the loop even if it's not needed for proper execution of the program.

I can only assume this is added because people copy and paste without understanding what's going on but it got me thinking if adding a delay could possible save power too.

If I have a loop that checks some inputs for changes could it be possible that adding a delay(10) saves some cycles and thus power?
No, it burns energy by doing nothing.

You definitively should send your MCU to sleep and wake up on events or use the watchdog timer to wake up after a predefined period of time.