A line of credit is interest only. You DO have to make payments though. If you borrow the max amount that you are allowed on your Line of Credit,, then the only way you can borrow again would be to pay down the amount owed. But yes, you can be foreclosed on if you do not make your interest payments.
A mortgage company CAN foreclose on a delinquent line of credit, but in most cases they generally won't because they have to payoff the entire first lien before they will get any money from a foreclosure sale when they are in a second lien position. The lien is however unlikely to ever go away and the balance can grow, but unless there is a lot of equity they generally won't execute on a foreclosure.
An equity line of credit is secured using the equity value of your home. You home becomes the collateral in the event of non-payment. Depending on the amount of the equity loan and applicable state and federal laws, the lender will have options such as putting a lien on your home to yes even foreclosure. That said, if you are going through a financially stressing time equity lenders are usually happy to assist you as long as you make the interest payments on the loan as Justin shared. As long as you do that it prevents the lender from foreclosing on your home. This also gives you time to seek out advice and search for options to resolve and relieve the financial stress.