I have a 10-year old HP workstation, worked great until it would not start up. The power came on the shell's button for 2 seconds, turned off, then shell button lit up again, and fans turned on. But computer did not boot up, no BIOS or anything. I got a new power supply and hard drive, but same thing happened. The last 2 components to try are the motherboard and CPU. If you were to guess, which one is more likely to need replacing? I'll plan to replace all the components (PSU, HDD, CPU, mobo) eventually, so it is not a critical question. I am just interested in guessing which part actually is necessary to replace, and only spend money on that for now. If you're concerned -- I got a new computer, and hooked up drive recovery kit to the old HDD. I was able to retrieve old files off it, so old data is not lost.
Check the condition of the thermal paste,between the cpu and heatsink. It drys and hardens with age. https://www.bestbuy.com/site/searchpage.jsp?id=pcat17071&st=thermal+paste
Could be a sign of "chip creep". Remove and then put back in any chips that sit in sockets. I.E., CPU, memory chips, etc. Could be something as simple as that.
Have you tried booting into settings mode? (On an HP usually by continuously pressing F10 while booting)? Disconnect any external speakers and listen for any beeps while booting. The number of beeps and duration can indicate what the problem is. Some computers also have one or two green/red LED lights whose blink sequence indicate what the problem is. Also, when was the last time you changed the CMOS battery?
A dead CMOS battery would not prevent a boot to the BIOS. It would simply not save any changes made to the BIOS.
I was looking on ebay for mobo / CPU replacements and removed the CPU fan -- I think this might be it. The thermal grease looks kind of spotty, feels gummy.
I forgot to mention -- one of the things I tried was I replaced the CMOS battery, hit the power button with everything unplugged to fully discharge, waited 5 minutes.
IMO, this is a waste of money and time. As I said, all the parts after 10 years will start to fail. Save your data off your hard drive, use a service like Dropbox in the future for all personal files and buy a new PC that fits your needs for the next 5 to 10 years. I built my last 9 computers over the last 18 years. I would not try and save parts that old.