Canon T3i with Blackmagic Design Intensity to record plays?
The problem is that the HDMI output on your T3i is not clean.  You would still have all the UI from the camera interface placed on it.  Higher end cameras (like the 5D mark iii) and some other makes have a clean output option, but I don't believe the T3i does.

You can use the third party MagicLantern firmware to force the interface clean, but at that point you can also have it automatically start a second recording after the first ends and bypass the 30 minute limit altogether.

As far as sensor overheating, it really depends on the camera model and the temperature in the environment you are shooting and how fast it dissipates waste heat.  MagicLantern generally displays an indicator of the temperature of the sensor with some color coding as to how hot it actually is.  You could try using ML and see how your camera handles it, but it may very well run too hot when trying to do two hours at a time.  You won't know for sure until you try though.

As others have noted, ML may or may not void your warantee.  It technically isn't a full third party firmware, but rather just runs on top of the underlying OS that is built in, but it can tweak settings that aren't normally tweaked internally.  There are plenty of cases of people getting warranty replacements on cameras that had had ML installed, but you should be aware that it is not a completely risk free process and in rare cases ML has caused cameras to become unresponsive (or even brick in exceptionally rare cases).