Does your processor fan have to work?

Do you mean does it have to run/spin?

Yes, it is very important that this fan works otherwise the core will get too hot. if it gets too hot, you risk damaging (melting) components. most computers have a sensor that will shut off the computer if it gets to hot, but to be on the safe side, i wouldn't use the computer until the fan is replaced.

However, there is the possibility that you are using a passive cooling setup if your computer is fairly old (more than 5 years) in which case, there shouldn't be a fan so its not a problem as long as it has been running fine up until now.

Or, you could have a water cooling setup but if you have to ask about the fan, you probably don't have it water cooled