MD-11 Pilot reported on take off an accidental gear retraction led to a tire failure alert and a return to the departure airport.

Narrative

While acting as Pilot Flying Aircraft X ZZZ to ZZZ1; during our takeoff roll and shortly before rotation speed a 'Tire failure' alert was triggered. As we continued our takeoff; once airborne the Pilot Monitoring inadvertently retracted the landing gear; however extended it again to remain compliant with company procedures. As we continued our climb; the Captain decided to assume aircraft controls and asked me to run the QRH procedures for TIRE FAIL. At the same time; we notified ATC about our situation; intentions to return to the airport and requested fuel dumping. Upon reaching ATC assigned altitude and vectors; we were authorized to start the fuel dump; which we performed as per the QRH's FUEL DUMP checklist. Before we landed on Runway XXR; we decided to [request priority handling] as a precaution and the landing was performed uneventfully. Upon exiting the runway; fire truck personnel indicated over the radio that all landing gear tires looked well; with the exception of a nose-wheel tire that seemed low. We then taxied to the gate uneventfully.[Identified] after V1; shortly before rotation speed. Although the tire pressure indications for one of the nose gear was deferred; the TIRE FAIL alert was not deactivated. Follow company procedures and ensure proper maintenance procedures for deferrals are performed.