I have a 5687SAX that would cease to respond to the remote after it had been on for awhile. I had started this subject earlier, and other owners with different models had said they also had that problem. Today, mine was fixed. It is the IR sensor module that is integral with the ON/OFF switch on the front bezel. Part # BP96-01792C.
When mine was not working, the technician sprayed some cooling gas onto the part and it began working again. There was no doubt this was the culprit!
Part exchange took about 15 minutes....remove front screen, disconnect wire to IR sensor, remove & replace sensor. then reassemble.
It would be a good idea to have a white image (I used Powerpoint to make a rectangle 1920*1080, white, saved as JPEG) to load into USB port after the work is completed to ensure there are no smudges, or foreign material on the inside of the screen.