Does the FT-817 have thermal protection?

The FT-817 doesn't have any thermal (overheat) protection per se:  Ideally the user would note that the radio is getting too hot during use. It isn't a good idea to run any radio at full power while obstructing its cooling:  Also, engaging in long-winded QSOs and using full power while the radio is enclosed in a carrying (restricting ventilation) case isn't a good idea. 
One slight amount of implicit thermal protection is the fact that MOSFETs (which are used in the '817's power amplifier) have the property of their gain reducing with increasing temperature.  Theoretically, this means that the output power will drop as they get warmer. 
In reality, this "thermal foldback" property will probably not prevent a negligent operator from overheating the radio and damaging it.  Many radios (the FT-817 included) have a feedback circuit used for controlling output power that will increase the drive power to the power amplifier to compensate for its drop in gain. 
In a nutshell, don't obstruct the flow of heat if you are going to run your '817 (or any other radio!) for long periods at high power.  Also, the FT-817 has a Time Out Timer (menu item 49) that can be used to help prevent an accidental unintended keydown (i.e. sitting on the microphone) as well as minimizing long-duration transmissions.