Tire flipping uses the legs, hips, and back. So it's the same muscles as squat and DL. However, the motor pattern for a good tire flip is not really the same as a squat or DL.
As long as you keep the flips light enough that they are more of an explosion rather than a grind, I don't think they will prevent you from progressing on the squat and dead. That said, I haven't heard of much carryover other than just the fact that being strong is important to a good squat and deadlift.