There is, perhaps a confusion in terminology.
In the US, a cpu cooler consists of a heat sink and a fan to cool the heat sink.
Some people call the whole combination a fan.
If you mean you want to remove the cooler from the cpu for whatever reason, the key is to run the cpu a bit first so it gets hot and softens the thermal paste so the cooler will not stick when you remove it.
Leave the cpu attached to the motherboard.
If you mean that the fan on the cooler is dirty, just clean it with canned air or a moist cleaning cloth.
No need to unmount the whole cooler.