I started writing this 2 years ago, got bored, and never finished. Posting it now just to get it out of my drafts.
If a superintelligent AI gains power over the physical world, it's very likely to want to expand its influence quickly. This is most obvious for a misaligned SAI such as a paperclip maximizer; if it has a utility function that rewards it for making as many paperclips in a short time frame, it will want to gain access to as large a portion of the universe as possible in a short time.
Perfectly aligned (or only slightly misaligned) SAIs will also want to expand quickly. Even setting aside questions of whether we should colonize the universe, any well-aligned SAI will have a goal of preserving humanity's existence, which means preventing all other civilizations from creating their own misaligned SAIs. The most straightforward way to do this would be to send out probes to other planets that ensure that those other civilizations do not create an SAI that poses a threat to humanity. The faster those probes can move, the better their chance of preventing the rise of SAI with conflicting goals.
If another civilization has already created an SAI with a conflicting goal and they come into contact, there are many ways that could go down, but in most scenarios it will be an advantage to control more of the universe than the other superintelligence does. This again suggests the desirability of developing maximally-fast ways to move across the universe.
FTL travel also increases the amount of the universe we have access to, potentially to an infinite degree. This means that any future civilization that wants to access more negentropy will want to use FTL.
Given the likelihood that superintelligence leads to fast expansion in some form, the question of whether faster-than-light travel is possible has implications for several current questions, and many more far-future ones. For example:
Several methods have been proposed that do not violate general relativity, like the Alcubierre drive, tachyons, and traversable wormholes. All of them tend to require positing the existence of types of particles that we have no particular evidence to believe exist, but some approaches might get around that, like this paper that talks about a construction of the Alcubierre drive with only positive energy densities.
If general relativity is even vaguely correct, any form of FTL information transmission, no matter how it's accomplished, would allow backwards-in-time communication. This would be pretty surprising, but not necessarily paradoxical.
One important implication that I believe this has is that if FTL is allowed at all, there's no "speed limit". It cannot be the case that, for example, traveling at 2c is possible, but traveling at 3c is impossible. To see this, imagine using your 2c method to send something back in time, then in the past do it again, etc. This gives you an arbitrary amount of time to travel at 2c towards wherever you wanted to end up. (This could be upper-bounded by extreme heat shortly after the big bang, so there might be a maximum distance you're able to travel this way.)
All of these would seem to point towards FTL being impossible, but given our poor understanding of the laws of physics, early universe, and anthropic philosophy, cannot entirely be ruled out.
Nothing can be "ruled out" 100%, but a lot would have to change for FTL travel to be possible. One thing that would have to go is Lorentz invariance. Which means all of current fundamental physics, including the standard model of Particle Physics, and the Standard model of Cosmology would have to be broken. While this is not out of the question at very high energies, much higher than what has been achieved in particle accelerators, or in any observed natural processes, it is certainly incompatible with anything we observed so far. There are plenty of open problems in fundamental physics, but it is not likely they would be resolved without understanding what happens at very high energies, far beyond those created in the heart of the supernovae explosions.
Lorentz invariance doesn't rule out FTL, it just makes it difficult to reconcile with time imposing a partial order and would require a rethink of strict causality. Alcubierre-like metrics and wormholes are fully compatible with Lorentz invariance, since they use general relativity as part of their base assumptions. Creating such structures poses unsolved and quite likely unsolvable difficulties, though.
Tachyons are also compatible with Lorentz invariance, but we don't have any quantum theory that works for them. There is the slight problem that no tachyons have ever been observed, and we don't have any reason to expect that they can exist. They also pose problems for causality.
Lorentz invariance does rule out crossing between disconnected components of the Lorentz group, at least classically, and thus FTL. Tachyons, if they were possible, would require a modification of Lorentz invariance to avoid traveling back in time, which is also prohibited in GR by the uniqueness of the metric.
Alcubierre drive is a slightly different beast. Beside needing negative energy, it has two other issues: the inside is causally disconnected from the outside and so there is no way to start or stop. Additionally, if you overcome this issue and manage to create an Alcubierre drive, you cannot go FTL outside the lightcone of the moment of its creation, though you potentially could travel FTL within the bounds of it. This is because any disturbance of a metric propagates at most at c. Sadly, I don't have an arxiv reference handy, I remember people publishing on this topic.
Wormholes are indeed within bounds of GR if one allows for negative energy, but they have a whole lot of other issues, one of which is that each traveler adds its mass to the entrance's mass and subtracts it from the exit's mass, so a lot of one-way travel would actually create an object with negative mass. There is also the issue pointed out by Novikov long ago, that wormholes tend to create a Cauchy horizon.