Since we now know these mugs are made from Polypropylene, I have been reading the Material Safety Data Sheet (MSDS) for this chemical here: Polypropylene MSDS
But let me summarize it for you for what you might like to know.
The melting point is 174 degrees C, which is about 345 degrees F. And it has a flash point around 300 deg C (572 deg F). If you heat it past its melting temperature, I am sure you won't be able to use it as a mug anymore. It would likely begin to melt before you reach that temperature, but in any case, it is very unlikely for anyone to actually push the mug to its flash point, which is the temperature at which it may spontaneous combust. This will degrade the material and make it potentially toxic.
But since our water boils at 100 deg C (212 deg F), with slight variations based on the ion contents of the water and elevation at which you are boiling water, it is unlikely that your beverage can reach this temperature. There are a couple of scenarios that I can think of where this might happen, ie, if you boil your coffee in a pressure cooker, but that just sounds silly.
Conclusion based on the melting point of the plastics: it shouldn't be a concern based on immediate temperature effects.
Even though the MSDS lists that it has no known chronic health adverse effects and that most skin contact is negligible in terms of health risks, I wouldn't want to be drinking hot things out of a plastic mug everyday. I migrated to stainless steel/ceramic/Pyrex for my materials of choice for hot-beverage vehicles.
Under the health risk category, there is an item listed for fine dust of this material and it may cause eye irritation and that you ought to avoid inhaling it. So if you purchased some of these, make sure that you wash them really good before using it, in case there are any fine particles left over from the manufacturing processes.
I have to point out that if the product undergoes thermal degradation, its byproducts will be simple hydrocarbon chemicals such as methane and ethane, propane, etc, but the list also includes carbon dioxide, carbon monoxide Acute exposure to these chemicals may give you symptoms including headache, nausea, irritation of the eyes, skin, and respiratory tract.
I do not know what temperature that the material may begin to degrade, but I will just say that I am done with drinking out of plastic for my hot drinks...