It might not be necessary to remove the spouse with Alzheimer's from the loan. However, it may be important to put the house in the healthy spouse's name to protect the home from Medicaid and possible liens. The house can probably be put into the healthy spouse's name without changing the names on the loan. Of course, someone still needs to pay the loan and the person paying likely will be the healthy spouse as time goes on since the ill spouse's income might get used up paying for care even if on Medicaid.
These are important questions and the answers can vary depending on where you live. I recommend you contact an elder law attorney before taking any action.