Its best to use a chinchilla comb because the hair is so thick. Chinchilla combs are made especially for their special fur. They have sharp points on each of the comb teeth to help get through the fur. Cat and dog combs are the closest thing but I think they would just pull out the fur (dead and live fur) rather than doing what you want them to do. If you don't want to spend the money for chinchilla combs (they do cost $25 or more a comb) try to get one that will only work on the surface. That way you can't do much damage. Sometimes one of those tint rollers made of masking tape will remove the hair. A finger nail file (emery board) rubbed back and forth through your chinchilla's hair will remove a lot of loose fur.
I have a few girls out there right now that look pretty terrible with fur sticking out all over the place!! Since my girls are pregnant I can't get them out to groom them. If there are just a few tufts here and there and I try to pull them out as best I can. But don't feel bad if you can't get them all. Eventually your chinchilla will get thought this prime and all of the old tufts will fall out and it will look fine again.