False.  Though a common lay myth, it is most likely untrue.  According to at the National Cancer Institute (NCI), a part of the National Institutes of Health, "(We) are not aware of any conclusive evidence linking the use of underarm antiperspirants or deodorants and the subsequent development of breast cancer. The U.S. Food and Drug Administration (FDA), which regulates food, cosmetics, medicines, and medical devices, also does not have any evidence or research data that ingredients in underarm antiperspirants or deodorants cause cancer." (posted 5/10/13 by @drsuzyyhall)