We know that taking photo of the sun can cause some degree of damage to the camera and our eyes. So, are there any damage to the camera (point and shoot / DSLR) when taking photo of the moon and/or stars at night? (Because the moon gets the light from the sun.)
The light from such objects is nowhere near bright enough to cause damage to the sensor, however using really long exposures to capture dim distant stars could damage the sensor by overheating.
Most modern DSLRs have heat sensors and cutoffs to prevent this, but if you're using an older camera and keeping the shutter open for hours with an external power source I imagine you could do some damage!
I'm not trying to be alarmist, but I would recommend a series of short exposures (less than one minute) with cooloff time in between for the best results. Multiple exposures can then be stacked in software to reduce noise. Unless you have a tracking mount you'll get star trails with longer exposures so that's another bonus of short exposures.
The moon is not anywhere close as bright as the sun (it's basically a giant diffuse, grey reflector), and so there is little risk. According to Wikipedia, the full moon at its brightest is about 400,000 times dimmer than the sun. That's 18 1/2 stops!
However, the moon does tend to get overexposed in photographs, since it's hard to fill the frame without a really long lens and space is really dark. As such it's best to shoot in manual mode or bring exposure compensation way down.
This is a repeat of
No, there cannot be any damage, not to the camera and not to your eyes either.
In the case of the moon and/or planets, there is not enough brightness to cause any damage, and the lens cannot increase the surface brightness. In the case of the stars, they are never more than a single point of light, so even with the largest telelens the brightness is too low to damage anything.