I believe an Owner's policy language provides that the coverage continues even after the Insured conveys title to the extent the Insured may have liability for warranties contained in the deed of conveyance or holds on interest in the property by virtue of holding a purchase money mortgage.I also believe a lien holder's interest, such as the OP's aunt, is considered to be "on the title" while not holding title.While a Notice of Sale is commonly used in NJ to notify the world of an upcoming sale of the property it is not ubiquitous since the Notice is not used in many states.

My aunt is in one and she is one of those people that tries to get all the renters and Airbnb people out.

I made friends with couple tenants, so did my grandma with her tenant of 15 years, so did my sister with her tenants, as well as my aunty who was a onsite property manager. 

On the dying aunt case, if you have full cooperation of all family members, and the aunt is alert enough to understand what she is doing, I would get a deed, going through the title company.On the second case, consult with the title company attorney, (not a title or escrow officer), to determine if a probate or a quiet title action is required to obtain good title. 

@Brandon Turner@Joshua Dorkin Also, how could he tap into the equity of the property his aunt is giving him, while mediating the risk of losing his aunt's residence?

My Aunt is moving out of the country and selling/gifting me her home in Yorba Linda for $800,000.

My wife's Aunt is a realtor , Investor she also advised against it. 

It's only the former generations that do not work (grandfather, father, uncle, aunt). I

I just moved to AZ, my aunt is a realtor here and uncle is a contractor.

My aunt is a real estate agent in Maine who got a call from someone who wanted to buy a house from them in Maine and needed to sell their home in Denver.