The existence of the IOLTA or IOTA is not attributable to state law, but to the rules of the state bar association or state supreme court, whatever organization it is that governs the practice of law in a particular state.
Banks are not the regulated party, lawyers are. If your bank does not want to offer them you do not have to; i.e. you do not answer to the bar association.
Generally, lawyers are required to open interest bearing accounts where client funds that are in small amounts or will be held for short periods of time are commingled.
Florida's rule is representative. These accounts are loss leaders to be sure. Banks accept them only because they want the rest of the firms's accounts. If you refuse the firm's IOLTA account, you will likely lose their other accounts. (Banks that hold only the firm's IOLTA are candidates for the remedial Banking 101 course.)
Whether they are or are not covered by deposit insurance is the lawyer's issue, not the bank's. (Yes, I too can conjure up the image of a lawyer who would say it was the bank's fault.) Regardless, the attorney is required to maintain accurate records of client ownership so I agree with John's analysis.
FDIC Deposit Insurance Seminars