if the processor is bad, it cannot process anything, so it cannot do anything like warn you the processor is bad. The CPU is needed to even POST because it does the processing.
If it is not starting though that can be many things.
It can be the BIOS doesnt support the processor even though the motherboard does. In that case you need to download the new firmware, put in an older processor that is supported, flash the BIOS and then put your new one in.
It could be that your RAM isn't supported. If the RAM isnt supported some motherboards beep some dont. Some error out, and never initialize the video.
Which brings another point, how are you testing nothing happens? Just by trying to watch the monitor?
Anyways, only way to test the processor is to put the processor in another known working, known stable system with an equal processor. If the system doesnt work then it is the processor. If it does work, then it is something else on the PC.