I am unsure, but whether a processor is off or on, I would believe that thet temperature still matters... though I can't see too much damage happening until above 70 Centegrade.
I'll give you an example:
I was recently talking to a mate in a computer store, he explained to me the silly things some people were doing when they tried to "home build" their desktop machines... he said he had met one guy, who when applying the thermal paste to the processor, spread it all over it, instead of just on the core.
What this cuased was when the pc was on, for the heat to be disipated accross the processor, instead of through the heat sink... which eventually caused a mild crack in the ceramic (or so I have been told, not too sure about the heat level required to crack cermic, but I believe it could be higher than 110C) Anyway... the just of the story is, when the temp got above 80C, the processor started to degrade and eventually refused to work.
I would imagine that whether the processor is on or not, temperature can still affect it, but I have seen processors survive 80C... though not recommended... they were not hurt by the temp.
It also really depends on your processor... some are of better quality and tolerance than others
tbh, I wouldn't worry too much as long as you don't run it as that could lead to an increase in temp.
thats processor wise... unsure about battery wise... you'd be better to check that out with some1 else
But thats my opinion