Can photographing the sun damage a camera sensor or other parts?

I would say it really depends on if you have a SLR, DSLR or P&S (Point-and-Shoot) - and maybe even possibly it more (or less) depends if the sensor is CCD or CMOS.

My own experiences says it doesn't occur with P&S cameras - ever. I have 4 cheap P&S (Canon PowerShot) cameras which I have used exclusively over the years for shooting time-lapse series (500 - 100k images per shoot) and all cameras have a minimum of 500k exposures according to Exif data. 2 of 4 have taken between 2 and 3 million images. Many of those sequences were shot during an entire day with complete and direct view into the sun (intentionally). Most of those were shot using a wide-angle or a 180-degree Raynox fish-eye adapter, though I would think that would compound the problem having more lens elements to refract the light?

Oddly enough, all 4 cameras have never exhibited any loss of quality or showed any signs of damage from the sun. However, one with the most exposures has a big purple splotch that sometimes appears in images off to one side, although it seems to be more connected to the LCD screen as when I wiggle the screen it usually goes away (for awhile) - I attribute it to falling off a rock onto other rocks and once falling 2m into the bathtub (which incidentally cracked the fish-eye, so now it rattles when you tip it any direction but amazingly still works fine ;).

I also have a Sony DSC-R1 (a high-end CMOS-based P&S camera) and have accumulated easily over 200k on that one, of which many times I spent very long periods of time shooting directly into the sun in an attempt at making Photoshop-like solar flares ;) Again here I was never able to find any noticeable deterioration or difference between the first images.

Then again, maybe it was all just luck that nothing got burned ;)