Under what circumstances, and as of what date, might I be required to hold a US passport in order to fly domestically within the United States?
A US passport is one of several documents that can be used to establish your identity at a TSA security checkpoint. An obvious way to fly without being required to have a US passport is to have any other document on the list.
People who do not have documents on the list may nonetheless fly by undergoing additional screening:
Forgot Your ID?
In the event you arrive at the airport without valid identification, because it is lost or at home, you may still be allowed to fly. The TSA officer may ask you to complete an identity verification process which includes collecting information such as your name, current address, and other personal information to confirm your identity. If your identity is confirmed, you will be allowed to enter the screening checkpoint. You may be subject to additional screening, to include a patdown and screening of carry-on property.
You will not be allowed to enter the security checkpoint if your identity cannot be confirmed, you chose to not provide proper identification or you decline to cooperate with the identity verification process.
TSA recommends that you arrive at least two hours in advance of your flight time.
You are probably asking in connection with Real ID. A number of states are not currently compliant with the Real ID requirements, although most of those states have an extension to the deadline. IDs issued by states that do not have an extension will not be accepted on or after January 22, 2018:
When will I need to change how I travel domestically?
Starting January 22, 2018, passengers who have driver’s licenses or identification cards issued by a state that is not compliant with REAL ID and that has not received an extension will need to show an alternative form of acceptable identification for domestic air travel.
Starting October 1, 2020, every traveler will need to present a REAL ID-compliant license or another acceptable form of identification for domestic air travel.
(It should be noted that some states issue enhanced driver's licenses, which can always be used even if the state's regular licenses are not compliant.)
If your only acceptable forms of ID are your driver's license (or other state-issued ID) and your passport, then the date on which you'll have to start using your passport to avoid the additional screening depends on which state you live in. There is a Department of Homeland Security map illustrating this.
A related question: