The deficiency portion of a home equity line of credit would generally be dischargeable, but the creditor would be able to take action against the house to collect on their debt. In general, debts secured by residential real property must be paid or the creditor can foreclose. If you have already had a foreclosure, or if you lose the house in your Chapter 7, then the remaining amount due is usually discharged. Sometimes, a creditor can bring a lawsuit to prevent the debt from being discharged, but they would be required to show something unusual about the debt or the debtors actions in the case.
All debt except for those that are deemed nondischargeable are discharged. However, this does not remove liens on your home. It only releases you from personal liability so that once the house is foreclosed on, you no longer have to worry about the debt. If you want to keep the house, you will have to pay off any mortgages and HELOCs attached to the house. If you plan to surrender the house, then you will be forever done with that debt. This is assuming that the lender has no valid reason to object to the discharge of the HELOC.