A home equity line of credit is a lien on the house. When the borrower dies, the debt doesn't die with them. The house passes to whoever the deed or the will says, and the balance on the line of credit goes right along with it, still attached to the property.
What usually happens first is that the lender freezes the line of credit. Most agreements say the draw period ends at the borrower's death, which means nobody is writing checks against whatever credit was still available. The balance already drawn stays, and it keeps accruing interest at the same rate it always did. Read the agreement, because the language varies by lender, but plan on the ability to draw stopping and the obligation to pay continuing.
Then somebody has to deal with it. The estate can pay it off out of other assets. The heirs can keep the house and keep making payments. They can refinance it into a loan of their own. Or they can sell, pay the balance at closing, and keep what's left. Federal law helps here. The Garn-St Germain Act keeps a lender from calling a residential loan due just because the property passed to a relative by inheritance, so an heir who wants to keep the house and keep paying generally can, without having to qualify for a brand new loan. Ask the lender in writing what they require and get their answer in writing too.
If you're running Dynamic Banking, this is a live issue rather than a hypothetical one, because the whole point is that the line of credit carries a balance a lot of the time. Somebody three years into chunking a mortgage might have $40,000 drawn on any given Tuesday. If they die that Tuesday, the family inherits a house, a mortgage, and $40,000 sitting on a line of credit they can't draw against to make the payments.
The fix isn't complicated and most people skip it anyway. Carry enough life insurance to clear the mortgage plus the working balance on the line of credit, and look at the number once a year, because the working balance moves. A death benefit pays fast, it isn't taxed as income to the beneficiary, and with a named beneficiary it doesn't wait for probate. An heir holding that check writes one payoff. An heir without it sells a house on somebody else's schedule. Running the two side by side, a line of credit on one side and a cash value policy on the other, is what I laid out in this one on carrying both at once.
Two more things to square away. Whoever handles your affairs needs to know the line of credit exists, which lender holds it, and roughly what's drawn on it. A frozen line of credit doesn't announce itself in the mail the way a mortgage statement does, and families find out about it from a title search weeks later. And if the house is held jointly with right of survivorship, the surviving owner takes the property and the lien together, immediately, with no probate at all. That's usually the outcome you want, and it's still a balance that somebody now owes.
Pull your most recent statement, write the lender name and the current balance on the same page where you keep the mortgage information, and tell the person who'll be handling things where that page lives. Then look hard at whether the death benefit you carry today covers the mortgage plus the working balance, or just the mortgage.