I'd say it depends on the reasons for and the length of your fast. If you are doing it for health reasons then I'd say yes, it hurts your fast. Not only is chewing-gum full of chemicals (like the excitotoxin aspartame) it also stimulates your digestion, so you won't be giving your stomach a rest. You will not obtain the same cleansing effects.
If you want to be healthy and get rid of the bad breathe try drinking chlorophyll or strained vegetable juice.
If your fast is for other reasons (spiritual perhaps) then perhaps you can chew gum. I'm not sure. Not the one to ask.