Will processor burn if fan stop working

Yes ! and sometimes very quickly.
Do Not operate your pc without a working cpu fan.

Replace the fan if it died. Will it boot after you replaced it?

Not necessariy. Newer processors simply throttle and/or shutdown. I know because it happened on my Q6600 and it would shutdown within 30 seconds. It might burn if there's no heat sink or if it came loose, but I doubt that would happen under normal conditions.

your cpu has a heat threshold and will shut itself down before it catches fire.

Modern processors will simply throttle back their speed if they get too hot. Older processors particularly AMD processors will burn out. Please specify the processor part number. On some motherboards the computer will not boot if the CPU fan is not spinning.

pjmelect is probably right. Does it beep at all on boot? That might be a sign of it detecting the CPU fan is not spinning and shutting down. Or, some other component on the motherboard may have died and killed the whole computer and the dead CPU fan is just a side effect.

for amd cpus yes they burn very quickly,but Intel cpus do not burn and after a certain temperature rise,they automatically turn off to safeguard themselves

I guess you never have seen a burnt cpu?

Dude that happened to my old computer...is it when you turn your computer on (by pressing that button to turn it on) it boots, but the monitor doesn't show anything...Try taping the cap lock n number lock in keyboard. If it lights up means something wrong with your monitor...if it doesn't mean something wrong with your processor(i think)...sometimes it cause your fan is too clogged up.

Are you sure your fan stopped? Cause sometimes it does what you said when its moving at a slow speed. Replace the fan if so...

I've seen an old MD CPU burn, but the heat sink was not correctly installed. I've never seen a Core 2 Duo or newer CPU burn. They throttle or shutdown before damage occurs.

Actually its a bit more common than expected..Burnt cpus.

Was that one running without a properly installed heat sink or a defective motherboard that failed to shutdown?

If that is an 1156 chip, then it was due to bad 'sockets' (LGA) when they first came out. A bunch failed, especially when OC'ed, by electrical (not heat) burns. That is what the left pic looks like.