I wouldn't chew gum during, or after, a fast. Most gum contains the artificial sweetener called aspartame. Aspartame is composed of phenylalanine, asparctic acid, and a methyl ester that is broken down by the body into formaldehyde. If you know anything about the practices of embalming, you know that formaldehyde is used in preservation of the body. This substance is obviously a toxin to the body. There have also been studies demonstrating that the isolated amino acid, phenylalanine, causes certain neurological damage, and even cancer in some cases.
The story behind how Aspartame was legalized and released into the food supply is truly a remarkable one, fraught with malpractice and chicanery. I would recommend checking out the documentary "Aspartame: Sweet Misery, A Poisoned World" It is available for free on google video and is definitely worth a view for those concerned about personal health.