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I understand that any faster than light travel can be seen as time travel, or breaking causality, in some other reference frame.
My question is, will this always be the case for every instance of faster than light travel? Are there ever any instances where causality would not be seen as broken for any observer? Could there be a mathematical rule where "as long as you are traveling in X way, causality will not be broken for any observer"?
Causality, in physics, is defined as the principle that nothing can influence events outside its light cone, a region of points in spacetime one could reach by traveling at or below the speed of light. Faster-than-light travel would allow you to influence events outside your light cone, so it automatically breaks causality.
It depends on what exactly you mean by "causality".
(1) If you mean that "the cause must always come before the effect", then no, there is no means of FTL that does not break causality for some observers. That is, given any event A (the spaceship leaves Earth) and another event B (the spaceship arrives at a distant star) for which the connection is faster than light, there is always some observer for whom event B (the arrival) happens before event A (the departure).
(2) If you mean "no time paradoxes", then there is a way to restrict FTL so that there are no "grandfather paradoxes" (closed timelike loops). That is to require that the FTL communication or travel always happen at a fixed speed relative to some frame, regardless of the initial (slower than light) speed of the spaceship. For example, the FTL speed may always be measured as 1000c relative to Earth, regardless of how the spaceship was moving before going FTL. This explicitly breaks the principle of relativity (the FTL works differently for observers moving relative to one another). Since it breaks Lorentz invariance there may well be implications for the laws of motion. So it's highly unlikely that any such FTL mechanism can exist in our universe.
Particles that accelerate past the speed of light are ruled out by special relativity but particles that have always traveled faster than light are not. These so called tachyons, if they exist, would need to have their interactions cut off from those in the rest of the universe. Any way for us to send or receive them would contradict causality as the previous answer points out.
But there are a couple other things that belong in a survey of faster than light proposals. Some physicists have been very interested in the possibility of closed timelike curves which seem to be allowed by what we know (so far) about general relativity. But here I would emphasize the keyword closed. As with the speed of light barrier, one cannot "enter" one of these curves. They describe an object which is in a perpetual cycle of causing its future self to cause its past self to cause its future self to, etc. A recent study on this has received some hype in the press but it is definitely not about time machines. It describes a communication protocol that can occur along a closed timelike curve and not lead to a grandfather paradox.
Finally, there are wormholes which lead to interesting things like Polchinski's paradox. Some people say wormholes might as well not exist because they would require huge amounts of energy. But that's a red herring because the same can be said about any macroscopic smoking gun of general relativity. Earth's gravitational field is weak enough that we can barely tell the difference between Newton and Einstein gravity. And yet $M_{\mathrm{Earth}}c^2$ is certainly larger than what any power plant can produce. Really, whether wormholes can exist depends on which energy conditions are obeyed by matter in this universe. In quantum field theory, which is the most fundamental description of matter we have right now, I'm aware of a few things that have been proven. The null energy condition does not hold but the averaged null energy condition does. As I mentioned in a previous answer, building on these results is an active area of research.
