As others have pointed out, it's not normal for a processor to start producing more heat than a heatsink/fan combo can dispel consistently, so it's most likely not the processor. Since the heatsink is getting hot, it's definitely conducting the heat from the processor. This indicates that the thermal paste is most likely not the problem either. Power supplies generate large amounts of heat, thus the reason that they had fans before even processors did. So I believe something else is going on here.
One thing I think people are missing is the fans. While the system is running, see if any of the fans seem sluggish. If one of the intake, exhaust, or CPU fans isn't running at full speed or is intermittently cutting out, that could cause the heat to not dissipate fast enough. This would cause overheating.
As others have mentioned, a good cleaning will help. If you can, remove the fan from the heatsink, blow all the dust out of both the heatsink and the fan itself and then mount the fan back.
However, if having the side of your case open helps cool things off, I'm inclined to think that the true problem is the case fans and not the CPU fan. It appears that your case has one 80mm fan on the side, one on the back, and one in the PSU. Make sure that each of these fans are properly running. Also, make sure that they are blowing the proper direction. The side panel fan should be sucking air in from the outside and blowing it toward the motherboard (intake fan) and the rear fans should both be blowing air out (exhaust fans).
If you can, purchase another 80mm fan and mount it in the front fan location. It won't do a lot of good since there's a lot of material blocking airflow for that location, but it will help introduce a new source of fresh cool air.
One last component that is often overlooked is the graphics card heatsink and fan. Pull the card out, and give it a thorough dusting. Make sure to blow all the gunk out by pointing the air directly between the fins. Lots of little dust bunnies gather there. With the case open and the system running, gently lean the case so you can look at the underside of the graphics card. While looking there, ensure that the fan on the card is spinning at a fast rate. If your graphics card fan stops, the card will quickly overheat and cause system failure.