A Home Equity Line Of Credit (HELOC-2nd Mortgage) is a lien against the property usually in second position to a primary loan. Since your house is paid off, it would then become the primary loan in first position to be paid off, if you were to sell the property.
If you become delinquent on the HELOC (all you are required to pay is the interest each month, usually, and perhaps very little of the balance). The terms of foreclosure due to not making you HELOC payments would be spelled out in the paperwork you would have signed at the time. Most likely if no payments on this type of loan are received by the mortgage company after a few months, then it would become at the very least a "term mortgage" or that you have given up the right to pay interest only, and that they could also start foreclosure proceedings against they property, especially now since the loan has become the primary lien and not a junior lien (no mortgage before it).
Usually when a home is in foreclosure and it has two loans (primary and junior -heloc) then the primary loan will take precendence to be paid during foreclosure and the junior lien would probably have to accept only 10% of the balance if the house is not sold for enough to cover both liens. The primary loan may accept 80% of the balance or a little less, but they will get most of their money - especially if there is mortgage insurance also involved, which helps the primary lender to recover up to 20% of the loan amount.
I'm not a loan officer, but I am a Realtor. I'm not sure exactly how all of the above works, but I am giving you the gist of what I believe to be what happens.
I would therefore say it is a recourse loan and not a NON-recourse loan. There are repercusions to not paying a HELOC loan, it is a lien against the property (usually a junior lien only to the primary loan) All other leins would be behind a HELOC loan (other than a primary mortgage would have first place, then the HELOC, then all others)