Chewing gum seems harmless, and there are harmless versions of it, however many of the mainstream chewing gum brands we find in grocery stores have chemical ingredients that are very concerning. Specifically, the carcinogenic preservative BHT and aspartame. Always check your gum ingredients for these two chemicals.
BHT is so controversial, that food giant General Mills recently removed it from their cereal.
Aspartame
It’s difficult to even put into words how destructive and insidious the diet soda ingredient aspartame is. A history lesson: In 1980 the FDA banned aspartame after it was confirmed (using 3 independent scientists) that it might induce brain tumors. LESS THAN A YEAR LATER (with no further research) the chemical was approved to be in our food by FDA Chief Arthur Hayes who left the agency 2 years later after a bribe scandal. Donald Rumsfeld and Monsanto made millions on this chemical, but thousands of people were reporting health problems.
So finally, in 
Ingredients aside, chewing gum too often is harmful because the chewing motion with the jaw releases enzymes from the pancreas and saliva which are produced in vein, thus leading to inefficient digestion.
Sources: Food And Drug Administration [Docket No. 75F-0355]; Testimony of Dr. Jacqueline Verrett, FDA Toxicologist before the U.S. Senate committee on Labor and Human Resources, regarding “NutraSweet Health and Safety Concerns.”(Nov 3, l987); http://www.fda.gov/…/Jan03/012203/02P-0317_emc-000199.txt; http://www.fda.gov/…/080102/98f-0052-her0001-vol5.txt ; http://www.repository.law.indiana.edu/ Other clinical/scientific reviews concluding aspartame-caused toxicity reactions: Blumenthal (1997),Drake (1986), Johns (1986), Lipton (1989), McCauliffe (1991),Novick (1985), Watts (1991), Walton (1986, 1988), and Wurtman (1985) ; https://cen.acs.org/articles/93/i8/General-Mills-Remove-Antioxidant-BHT.html; https://www.c-span.org/video/?1050-1/nutrasweet-safety-concerns

 
					