coppermine doesnt have much headroom above 1GHz. intel released 1.13GHz coppermine parts which were defective, which kinda speaks for the coppermine's headroom by itself.
i believe intel also released a tualatin core p3 1GHz. if you have one of these you have much more headroom, but i doubt you have one.
p3's ran cool compared to todays chips or the amd chips of that day, staying below 60 C is advisable, as with any chip. however the p3 has an integrated theermal diode which will shut down operation and load before it gets hot enough to damage itself.