Short answer: nobody knows for sure.

A French epidemiological study released in 2000 indicates that people who drink water that contains greater than 0.1mg/litre of Aluminum are at greater risk of contracting Alzheimers. The amount one gets in antiperspirant through the skin would be much much smaller compared to this. Aluminum-based antiperspirants contain 1 - 6 mg of Aluminum/litre. The amount you use is very small however. If you use a spray, some could get into your lungs or gut via inhalation of the aerosol.

One has to remember that aluminum is the third most abundant element on Earth and so we are all exposed to some extent. You get it in air, water, and food. The amount you get from antiperspirant is probably extremely small compared to the aluminum you simply eat. Studies on mouse skin show that aluminum is absorbed and concentrated in the brain. However, mouse skin is quite different from human skin. Mice do not have sweat glands for instance. Nevertheless, it's probably better not to use aluminum-based antiperspirant if you can avoid it. If you must, use a solid, or roll-on, not a spray, and only use it when absolutely necessary. In the absence of any solid scientific evidence it might be better to err on the side of caution.

About four years ago, a widely circulated email claimed that using antiperspirant deodorants prevent the lymph glands near the breast from excreting toxins through perspiration which can lead to breast cancer. There is no medical evidence to support this claim. It was a hoax.

References:

French paper: V. Rondeau et al., Relationship between aluminum concentration in drinking water and Alzheimer’s disease: an 8-year follow-up study, Am J Epidemiol 152 (2000), 59–66

A review of Alzheimers and Aluminum can be found at http://iospress.metapress.com/content/km5hy6bgy5hx3dt8/ with lots of references at the end for further reading, but it costs money to download.

The hoax about aluminum in antiperspirant causing breast cancer is covered in many places. It's amazing to me that hundreds of companies still use it to sell "herbal" antiperspirants. I'm not saying there's anything wrong with herbal antiperspirants. Maybe they are excellent, but the sellers discredit themselves by using totally bogus information to market their products. Look at

http://cis.nci.nih.gov/fact/3_66.htm

http://www.snopes.com/toxins/breast.htm