It sounds like the crash is coming from accessing a bad sector of RAM. Or it could be a bad sector on the hard drive. You'll want to check that the RAM is running at the correct clockspeed.
If you RAM is just halting and the PC is restarting, I highly doubt it will cause any damage. But if the issue is something more nefarious like the power supply/source or motherboard capacitor, then that could be a dig deal.
The first thing I would do is check to see if the problem persists with only one RAM DIMM. Yes, you will probably not be able to do much of anything on your PC while you do this, which sort of negates the test, but not really. Check each one individually to see if it is indeed isolated to a particular DIMM.
I would also check the manual for your motherboard and make sure they are indeed installed per it's recommendation. Not all motherboards arrange their RAM DIMM slots the same.
Out of curiosity, what brand of memory is it?