Home equity lines of credit are revolving credit lines, so lenders will most likely run a hard inquiry of your credit score to determine your eligibility, which could drop your total by a few points. Applying for a HELOC with multiple lenders doesn’t hurt your score more than applying with only one.
Being overdue on your HELOC payments or missing them completely will have the greatest impact on your credit score. Also, though it may seem counterintuitive, using your home and thus risking damage to it is another danger to your repayment, as your home is the collateral for a HELOC.