Assuming the most obvious obstacles were solved (rigidity, how you tug on a frictionless string [great point mike] etc), and you had the perfect string for this

Assuming the most obvious obstacles were solved (rigidity, how you tug on a frictionless string [great point mike] etc), and you had the perfect string for this experiment: Nothing would be moving faster than the speed of light. Every single inch of the string would be moving at the same speed, which would be a lot slower than the speed of light. But every inch would move in unison. I don't see how the "faster than the speed of light" thing has any bearing. That said, it's still a physically impossible experiment.