Your home equity line of credit would transfer to whatever bank made the purchase of your bank. In the event that no other bank purchased your bank, and the FDIC simply closed them without finding a suitable buyer, your HELOC would be sold to another bank.

When the FDIC seizes a bank, if there is no purchaser, they will begin to sell that banks assets off to other banks or lenders. Your HELOC is an asset to your bank, as it generates them income (the interest you pay). There is no need to draw the available balance on your HELOC, as it should still be available to you with the new lender.

Your terms can only be changed if it was written in your initial agreement, and in that case you will be notified of a change. They will not change automatically simply because your loan was sold to another bank.

As for payment schedule, it would remain the same until you were notified of a change. When an FDIC seizure takes place, the FDIC continues to operate the bank until it is unwound, so you would operate as normal, until notified of any change in who you make your payment to. That would be the only change, date of payment, etc would not change as that is again governed by your initial loan agreement.

If you're asking if you are released from any contractual obligation on your HELOC if your bank closes, then no. More than likely your HELOC will be sold to another lending institution or if the bank gets bought it will transfer to the new bank.  I recommend you keep making your payments on your HELOC unless you can prove you were a victim of Predatory Lending or Fraud.