Can people die from alzheimer's disease?

Yes. In the late stages of alzheimer's disease the body's vital functions break down. The body "forgets" how to swallow, the muscles grow rigid. The person starts to aspirate, they develop pneumonias, pressure ulcers and other infections that eventually kills them. It's important for patients and their families to talk with their doctor about an advanced directive before that happens.
Yes. Alzheimer's is the 6th leading cause of death. The most likely final cause of death when one has dementia is pneumonia.