When we see a star from the earth that is 5 light years away, we are essentially looking at light emitted from the star 5 years ago. What if you were to instantly reach near the star, either by teleportation or Alcubierre drive, are you traveling back in time 5 years?
Well not sure teleporting requires an answer. There is not such a physics. However assuming the impossible I do not think we really need relativity SR in a technical sense. If I teleport instantaneously myself I would appear on the star as it will be seen from earth in 5 years. As such, in the earth calendar, I disappear for 5 y and "in a way" I move in the earth future as for an earth observer it will take 5 y to see me again. But he/she will see me as I was during my absence, ie his past. As far the star is concerned, I simply materialize there. The point is that from there, I will see my last 5 y on earth as a recorded movie. The same is for a star resident watching the movie with me. He see me before leaving so I wasn't able to give to him any sensible information. Even a pre alert of my arrival wouldn't alter anything on the succession of events. This does not qualify as moving or moving information through time, especially not backwards. At the end, the fictional teleporting seems to lead to less mind blowing results than a technological difficult but in principle possible travel at a relativistic speed. There is even no room for time dilation and it seems to me that the famous twins can use teleporting and re-meet with no even (apparent) paradox.
Edit. Please note that the scenario can be made fully realistic (well almost, but because of merely technological limits) by replacing
earth and star by widely separated islands;
sound as the only way to communicate, such as big detonation while being away and whispering while sitting on the same rock;
a really supersonic vehicle for moving between the two islands, so that the time of the person travel can be taken to be negligible as compared to the time at sound speed.
Nothing can alter a cause effect succession of events even considering a third observer passing by or through the islands even on board a similar vehicle.