It depends on a number of factors, including how you are listed on the account. For example, if you are listed as a joint account holder, then yes you will have tax consequences. It also matters how the last will and testament is set up, if they even have one.
Generally, when a person dies all of their assets get transferred into an estate. The estate then pays any and all taxes that may be due and what's left is distributed according to the will. If no will, then according to state law.
I would *HIGHLY* recommend you seek the services of an elder law attorney *AND* a CPA for this. Laws differ state to state and what matters to you is the laws of "your" state as well as the state the other person resides in, if different from you. It can get complicated.