This was the question posed and answered in a recent Legal Interpretation issued by the FAA's Office of Chief Counsel (AGC). The Interpretation specifically answered the question "whether a chief pilot who no longer holds a first-or second-class medical certificate but holds a third-class medical certificate and is qualified to serve as pilot in command (PIC) in at least one aircraft used in the certificate holder's operation may continue to hold the chief pilot position."
Under 14 C.F.R. § 119.71 the chief pilot of a Part 121 or Part 135 air carrier must hold either an airline transport pilot (ATP) certificate or a commercial pilot certificate. And as we know, under 14 C.F.R. § 61.23 a person must hold a first-class medical certificate to exercise PIC privileges of an ATP certificate and a second-class medical certificate to exercise privileges of a commercial pilot certificate. The Interpretation notes that "§ 119.71(c) and (d) only require the chief pilot to be qualified to serve as PIC in at least one aircraft in the certificate holder's operation." However, Section 119.71 does not specify that the chief pilot must be qualified to serve as pilot in command "in Part 121/135 operations."
As a result, the Interpretation concludes as long as the person (a) continues to hold either an ATP or commercial pilot certificate with appropriate ratings, (b) is qualified to serve as PIC in at least one aircraft used in the certificate holder's operation (which can include just Part 91 operations), and (c) has satisfactory experience (or has been granted a deviation from the experience requirements), then he or she may serve as chief pilot for the air carrier while holding only a third-class medical certificate.
So, in addition to holding at least a third-class medical certificate, the key issue for the chief pilot will be that he or she is qualified to act as PIC in at least one of the aircraft that the carrier is authorized to operate under its certificate. And this qualification can be limited to Part 91 operations rather than Part 121/135 operations.
As a practical matter I think this Interpretation probably has limited impact on most carriers since they typically expect the individual designated as chief pilot to also act as PIC in Part 121 or Part 135 flight operations. But where the chief pilot is not expected to act as PIC in Part 121/235 flight operations, this Interpretation does provide some flexibility to a carrier considering an individual without a first or second class medical certificate for the chief pilot position.