My father is suffering from dementia. As a result, I must cash his monthly social security check and use the proceeds for his care. What are the resulting tax consequences?

Your father's social security benefits aren't taxable to you. In determining whether you provided over one-half of your father's support in order to claim him as your dependent, you should consider the benefits as funds your father paid for his own support.  

See Publication 501, Dependents, Standard Deduction and Filing Information for additional information.