PA44 Flight Instructor reported the student lost directional control on takeoff following a long landing touch and go; and the aircraft departed the runway after a tire failed.

Narrative

At ZZZ student was demonstrating touch and goes. Runway XX was in use by multiple aircraft when we arrived; even though there was a tailwind. The student landed well over a thousand feet long from a landing; likely due to excess airspeed but also the tailwind making us more unstable. Upon touching down; the student began to apply full power to commence the takeoff; prior to retracting flaps to 0. Flaps were at 25 which is not approved for takeoff. About 2000 feet remained of runway distance. Because of limited distance and incorrect flap configuration; instructor called out REJECT REJECT REJECT. Student did not reduce power; or close both throttles to idle. Student applied max braking with power still in which resulted in the LH tire popping. Incorrect rejection technique; as power idle should be first not last. Student was struggling to stay on centerline after tire malfunction; in which I called for controls of the aircraft. Applying full right rudder and right brake was not enough to keep us on the runway. The aircraft veered to the left and into the dirt next to the runway. No damage to the aircraft other than popped tire. We announced to ZZZ our position and then shut down the plane to assess damage. Called Airport Manager to tow the plane to the ramp.