In order for an idea to lead to a logical contradiction, there needs to be something that it contradicts. Since the idea of faster than light travel is logically conceivable, it has no self-contradiction, and therefore there is no a priori contradiction. Thus, we have the following answer.
Question: Does faster than light travel/communication lead to a logical contradiction?
Answer: No, because it is not a self-contradictory idea.
The only other way for us to reach a contradiction is for faster than light travel/communication to contradict something else. The obvious candidate is special relativity. So now we reduce our question to, does faster than light travel/communication violate special relativity? As I explain below, faster than light travel/communication does not contradict special relativity. Thus, we have the following answer.
Question: Does faster than light travel/communication contradict special relativity?
Answer: No, but there are interesting consequences here that make people think it is unlikely. Faster than light travel/communication implies either (1) travel/communication to the past is possible OR (2) special relativity is incorrect. Since people think special relativity describes the world and since people think travel/communication to the past is unlikely to be possible, it follows that people think faster than light travel/communication is unlikely. (And if you think special relativity is correct and travel/communication to the past is impossible, then you would logically conclude that faster than light travel/communication is impossible.)
Here I explain the conclusion of the above answer.
Two events $(t_{1}, x_{1}, y_{1}, z_{1})$ and $(t_{2}, x_{2}, y_{2}, z_{2})$ are said to be time-like separated if
$$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} < 0, $$
they are said to be light-like separated if
$$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} = 0, $$
and they are said to be space-like separated if
$$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} > 0. $$
Two events that are time-like or light-like separated are those for which you could go from one to the other in less than or equal to the speed of light. Space-like separated events are those that are too distant compared to the time you would need to traverse to get from one to the other.
The main premise of special relativity is that the laws of physics are invariant under Lorentz transformations. In the passive sense, Lorentz transformations are simply a change of coordinates. What this means is that if $K$ is one inertial coordinate system, and $L(K)$ is another coordinate system obtained by a Lorentz transformation $L$, the fundamental equations of motion should be of the same form regardless whether they are written in terms of coordinates $K$ or coordinates $L(K)$.
To give a related example to make this more understandable, consider taking an inertial coordinate system $K$ and consider rotating the coordinates any angle $\theta$ about the $z$-axis to get new coordinates $R(K)$. This is also a change of coordinates, and because the universe doesn't have any "intrinsically preferred direction," it doesn't matter whether we've written the laws of physics with respect to $K$ or $R(K)$. Hence the laws of physics are rotationally invariant.
To understand Lorentz transformations, it might be better to first think about Galilean transformations. A Galilean transformation is simply a change in the velocity of the coordinates (so if you have an inertial coordinate system and another inertial coordinate system passes by at constant velocity, the two coordinate systems differ by a Galilean transformation), and the classical notion that physics is invariant under Galilean transformations is the idea that there is no meaning to absolute speed.
A Lorentz transformation is similar to a Galilean transformation, except it mixes time and space coordinates. Einstein found out that, upon closer examination, physics is not Galilean invariant but Lorentz invariant. The invariance under partial mixing of time and space coordinates leads to time dilation, Lorentz contraction, and the invariance of the speed of light, but most of all, it leads to the idea that different inertial coordinate systems will disagree on the ordering of space-like separated events.
If events $A$ and $B$ are space-like separated, one coordinate system might say $t_{A} > t_{B}$, but another coordinate system might say $t_{A} < t_{B}$. This is not a problem, however, because no communication can occur between space-like separated events. Also, time-like and light-like separated events never change order. Thus, causality is always preserved.
Now the problem with faster than light communication (of any kind) is that it involves sending signals between space-like separated events. Let's say $A$ and $B$ are space-like separated, and I send a signal from $A$ to $B$. In one inertial coordinate system, $t_{A} < t_{B}$ and I sent a signal from location $A$ to location $B$ in a certain time period. But in another inertial coordinate system, $t_{A} > t_{B}$, and I sent a signal from location $A$ to location $B$ to the past.
But if sending signals to the past is possible in one inertial coordinate system, then by Lorentz invariance it must be possible to do this in any inertial coordinate system. Thus, by performing the same thing I did at event $A$, a person at event $B$ can send that signal back to my past self, and essentially a form of time travel is achieved, leading to the grandfather paradox.
Now the grandfather paradox is not a contradiction by itself, because it is possible that we are deterministically fated to avoid creating various paradoxes in our timeline, but the whole idea of sending signals to the past is too much to entertain.
If signaling at faster than the speed of light is possible, there is another possibility, which is that Lorentz invariance is broken, and that there really is an absolute coordinate system after all. If you go back through my reasoning, you would find that Lorentz invariance was the key component to $(1)$ changing the ordering of events $A$ and $B$, and $(2)$ generalizing "sending a signal back in time according to one coordinate system" to "sending a signal back in time in any coordinate system." If laws of physics ultimately depend on an absolute coordinate system, then it is possible to send signals faster than light according to that one coordinate system without any backwards time traveling.
So to summarize, if faster than light communication existed, then either special relativity is incorrect and there is an absolute reference frame, or signaling into the past is possible. Since we have no reason to think that special relativity is incorrect and no reason to think that signaling into the past is possible, we are left with the conclusion that faster than light communication is probably impossible.
Ultimately, it depends on what empirical evidence will show us in the future. Any one of the possibilities I mentioned is possible (with no logical contradictions), but until then, not much more can be said.
