The entire "FTL implies time travel" meme has to do with what's sometimes called "failure of simultaneity at a distance". In addition to the effects that pop treatments of relativity mention (that is, time dilation and length contraction), relativity proposes that the definition of "right now" is also different, depending on which observer's coordinate system you use. This is an effect much like the revolution of deciding that the direction "up" wasn't the same everywhere, but varied from place to place on earth. With relativity, the revolutionary notion is that the direction "futureward" (or "now-ward") isn't the same everywhere, and varies with velocity.
We can describe this effect by idealizing FTL to be "instantaneous", and describing how the more familar time dilation implies this effect. But remember, the same points apply to any FTL speed, you just have more messy arithmetic to grind through.
Consider a duel with tachyon pistols. Two duelists, A and B, are to stand back to back, then start out at 0.866 lightspeed for 8 seconds, turn, and fire. Tachyon pistol rounds move so fast, they are instantaneous for all practical purposes.
So, the duelists both set out --- at 0.866 lightspeed each relative to the other, so that the time dilation factor is 2 between them. Duelist A counts off 8 lightseconds, turns, and fires. Now, according to A (since in relativity all inertial frames are equally valid) B's the one who's moving, so B's clock is ticking at half-speed. Thus, the tachyon round hits B in the back as B's clock ticks 4 seconds.
Now B (according to relativity) has every right to consider A as moving, and thus, A is the one with the slowed clock. So, as B is hit in the back at tick 4, in outrage at A's firing before 8 seconds are up, B manages to turn and fire before being overcome by his fatal wound. And since in B's frame of reference it's A's clock that ticks slow, B's round hits A, striking A dead instantly, at A's second tick; a full six seconds before A fired the original round. A classic grandfather paradox.
Note, this is NOT a matter of when light gets to an observer, it is NOT an optical illusion. It is due to the fact that, in SR, the question of what occurs at the "same time as" something else is observer dependent.
As A fired that first show at tick 8, the bullet effectively teleported from A's gun to B's back instantly --- instantly according to A. But for B, who was moving at 0.866 lightspeed WRT A, B was hit in the back by the bullet 4 seconds BEFORE the bullet was fired. And again note, this is NOT due to the optical illusion of lightspeed delay in viewing A's turn-and-shoot; the light form that event wouldn't reach B until MUCH later, not tick 4.
Here's a spacetime diagram of a referee (O3) and two duelists (O1 and O2). Space is up/down, time is left/right, in the diagram.
The paths O1 and O2 take through spacetime are the colorcoded arrowed lines. The events in spacetime that each considers "simultaneous" and "8 seconds after the start" are along the thinner, non-arrowed colorcoded lines. So we see that, each of the three observers thinks the other two have slow clocks, and that if we are allowed to move faster than a lightcone, we'll end up going "pastward" in somebody's reckoning.
We can trace out the shots on this diagram, also. Step through it one more time: if green shoots at 8 seconds out, the shot will go along the green "line of simultaneity", and hit blue at 4 seconds elapsed. If blue returns fire from there, it will return along a line paralel to the blue "line of simultaneity", and catch green napping at 2 seconds elapsed.
You can see more about spacetime diagrams, and more about the notion of "lines of simultaneity" at
Finally, FTL still can can bite you in non-instantaneous cases; where we're only going a "little bit" faster than light.
If you warp out, go to Tau Ceti, then with normal reaction engines accelerate away from earth, warp out again to go back to earth, you will indeed get back before you left. (Presuming that the real-space delta-v before the warp/hyperdrive/tachyon-watziz trips was "large enough"... there are formulas for such things in the textbooks).