I think it would depend on the cheese.
For a reasonably hard cheese, like cheddar, I have done it, and never gotten sick (your results may vary).
If it's pre-grated, then I would not touch it anymore.
For a softer cheese, like a brie, I would not risk it.
cooking.stackexchange.com/users/260/Nick
As Nick said, I wouldn't recommend it for soft cheese...
the process of cutting off the mold can push some nasty bacteria into the inside.
I've also never got sick from doing this.
Now if you're talking about some piece of cheese that has questionable provenance (been in your student accommodation fridge for 6 months)...
well that may be different :)
cooking.stackexchange.com/users/335/octonion
Solid blocks of cheese, such as cheddar, will only have the mold on the exterior so cutting it off should be fine.
For soft cheeses such as Brie all you need to do is cut far enough back that you're not cutting through the mold itself.
Brie is more likely to develop an ammoniated smell before the mold gets too extreme and if this is the case cutting the mold isn't going to help, the cheese has continued to deteriorate on the inside already, simply throw it out.
cooking.stackexchange.com/users/426/Darin Sehnert
Hard cheese, yes, I'm still here.
Soft cheese, no, it travels through the interior of the cheese too quickly.
Also, if it smells odd after cutting the mould off, I'll throw the whole lot away.
I don't like to be in doubt about whether my food's going to make me sick or not.
cooking.stackexchange.com/users/539/Matthew Walton
Harold McGee (yes I know,
I quote him a lot)
suggests that molds can penetrate into cheese far more than just the surface contamination.
cooking.stackexchange.com/users/None/
As has been mentioned some types of cheese are deliberately infected with mold as part of the maturation process (for example Blue Stilton) and are perfectly safe to eat.
Eating other types of cheese which have had mold growing on it is a bad idea.
This type of mold produces toxins which spread throughout the cheese.
Cutting off the mold will make no difference.
I was told this by a food scientist.
cooking.stackexchange.com/users/58246/Chris Brown
Whenever my cheese gets too old and has a bit of blue/white mold on the side, I'm not sure whether I should throw it completely out or not. Is it safe to eat if I cut off the edge? Should I cut a bit off of all the sides just to be safe?  I'm particularly interested in the answer for cheddar cheese, but answers about other cheese are welcome.
Is it safe to eat moldy cheese if you slice off the edges?
cooking.stackexchange.com
<food-safety><cheese><mold>
cooking.stackexchange.com/questions/1230
cooking.stackexchange.com/users/319/mouche
Solid cheeses like cheddar can have external mould that does not spread inside and can be safely cut away. Softer cheeses like Brie are more risky. For hard cheese, you should be fine to cut off the moldy part and eat the rest.
For hard cheese, you should be fine to cut off the moldy part and eat the rest.