If you are a nonimmigrant traveling within the US, when going through security you will need to present government-issued identification to the TSA. This ID does not need to be a passport or any immigration document. A valid state driver’s license is sufficient. If you present a passport, and the TSA agent questions the validity of your immigration status, this could lead to a delay and possible reporting to the Customs and Border Patrol. Even if you are in valid nonimmigrant status, this delay could cause you to miss your flight.
When possible, avoid presenting a foreign passport as your preferred form of ID.