In general, no, though what you should be looking at is load, but temperatures. You might want to simulate the load (perhaps with superpi on windows, prime95 on linux and windows or some other stress test application, or your actual application) and run the process until the temperature stabilises -similar to what an overclocker would do.
You want this temperature to be under the maximum rated temperature for that processor -look up the datasheet for that. Then see if your cooling solution is adequate and presumably adjust it as needed. I can't suggest monitoring software, unfortunately without knowing an OS - the usual suspects such as nagios may work, but for a initial 'burn in', something like lmsensors or pcwizard may do for a single system.
Most modern processors will simply throttle down if they overheat - intel has a page on this. It would be horrible for your application if performance is critical, but its more likely the system will shut down or act strange rather than burn itself out. Thats a sign the processor is overtaxed, and bad things are happening - review your cooling choices in that case.