Not likely a CPU issue, but could be a bad motherboard. CPUs don't fail often unless you have been running it too hot for a while .
Need more details about the system and exact issue and what you tried. You said "to reduce heat", so was the system overheating before, or did you just assume it was overheating because it was turning off?
If fan is not working and it was overheating, but not booting now, then yes the CPU or some other component may be dead. CPU is soldered to the motherboard in many laptops so really the only thing you can try is a new motherboard, or replace the laptop with the same model and swap the drive into that.