Send it in to Canon, and have them clean and check it out. Since it is used, it is probably a good idea, anyway.
Most any DSLR sensor will eventually overheat after extended use in video mode. Most of the time Live View mode will shut itself off after a minute or so. But, I would think you should be able to get more than a few minutes of video from a 5D3.
Do you see a difference if you are shooting indoors in an air conditioned facility versus outdoors in high temperatures?
You may have a temperature sensor issue that is shutting down the camera sooner than needed but a DSLR is still primarily a still image camera with some ability to shoot video. The 1DX series is probably the best of the Canon DSLR bodies at standing up to video shooting with its large body and heat pipe cooling design that dumps heat to the battery compartment but it is still very much constrained by the need to make a DSLR weather and dust resistant rather than providing optimized cooling for video mode.
I picked up a Canon XF-400 camcorder for doing sports video work. Drawbacks are I can't use my collection of EF lenses for it and it is definitely not weather resistant like a 1 series body but it shoots continuous video without thermal concerns. It has forced air cooling which by default runs continuously but can be set to temperature activated mode to conserve battery life. That same fan will pull moisture and dust into the body which would be death to a typical DSLR.
In direct sunlight these black camera bodies soak up a lot of heat so if ambient temperature is high I can understand it overheating even in standard still image mode. Its only way of dumping internal heat is via transfer through the case and when the case temperature goes high from direct sunlight then the internal temperature is going to quickly reach an extreme temperature range.