I may not have been clear but what I mean is that an ansible can allow any two other observers (one on each side) to communicate FTL in a flat space-time and these observers do not need to be at rest (even if the endpoints of your ansible must be at rest) and this is all that's needed to run into problems. Also, the speed of the observers does not necessarily need to be close to the speed of light and the farther away the two observers are the lower the minimum necessary speed for causing violations. A more concrete example:
Let's say observers A and B are at point (0,0,0) and C and D are at (1000ly,0,0). A and C are at rest relative to each other and have an ansible connection between them that will break if they are not at rest anymore. B and D are not in anyway restricted so they can travel and any slower-than-light speed that they want. So, B starts going at 0.5c relative to A and to the left of A and and D starts going at 0.5c relative to C and to the right of C. Thus B and D are going at a significant speed in each other's frame. A sends a radio message to B, which immediately relays it through the ansible to C, which sends it via radio message to D. Then it goes the other way from D to C by radio, through the ansible to B and by radio from B to A. So, what you have is A and D communicating much faster than light, while also moving thus easily allowing them to send messages into their past. In this case the fact that the ansible only works if the two endpoints are at rest relative to each other does not help you unless you also make up something that prevents A and C from communicating with nearby observers.
Another example with a somewhat different setup but still exploiting the same problem is shown here, with pretty space-time diagrams: http://www.theculture.org/rich/sharpblue/archives/000089.html. In this case you have A and B with an ansible and C and D with a separate ansible. A and B are at rest relative to each other. And so are C and D. However A and B are moving relative to C and D and are able to communicate by regular methods, which violates causality.