Psychedelic drugs actually do not increase chances of developing Alzheimer's at all. In fact, there are studies being done that show that these drugs can decrease the chances and the effects of Alzheimer's.
This doesn't mean a dementia patient should run off and take large quanties of any hallucinogen, however there is potential to use this as a cure. Before giving any drug to your grandpa on the sly, understand that in all of these studies, microdoses are being used.
No, taking psychedelics does not increase your chance of developing Alzheimer's disease. Infact, some psychedelics are being explored as treatments for Alzheimer's disease and other types of dementia. Psychedelics such as mushrooms are in the experimental stages in order to determine the benefits and any potential side effects of taking the drug. It is likely that if a psychedelic such as mushrooms were to be used a treatment for Alzheimer's disease, there would be alterations to the drug in order to make it more pallatable for most patients.
This does not mean that people with Alzheimer's or other dimentias should experiment with psychedelics on theirs own. Psychedelics can be very dangerous to someone with dimentia if they are not in a safe place.