From everything I'm reading allowing an HSA to be overdrawn is a bad bad idea. It ceases to be an HSA as of the first day of the year of the od, if the irs catches them they can be accessed 10% penalty, and they owe the regular taxes. My question is: if the customer is od and cover the od the next day, Would the bank still be liable for not immediately withholding and remitting the appropriate taxes?
Also, would it be a good idea for the bank to ask the customer to close this hsa or to transfer the money into a new hsa? wonder how the irs would look at this?

No overdrafts, period. The day the bank pays an overdraft on an HSA they have breached their fiduciary responsibily as the custodian and would likely be held liable for all associated taxes, expenses, fines and penalties. It is a prohibited transaction and it would be the bank's fault and not the customer's.

Right now we don't have an HSAs, but I can forsee the day when we might. Would the comments posted imply that an ATM or Debit card should/would/must NOT be given for an HSA, as the use of one might cause an overdraft that we can't "return"?

I would not advocate issuing an ATM or debit card to an HSA customer unless you're dealing with online real-time updating. Even then, a debit card could create an overdraft that you cannot return.