From a physics perspective, it is mathematically possible, as shown by Einstein's paper on the tachyon anti-telephone. Einsteinian Relativity shows that, if FTL motion occurred, where one could either communicate information or physically travel FTL, in some frames of reference one would observe the signal's reception as happening before the signal was sent. A clever but simple arrangement of such tachyon antitelephones would allow one to receive a message before they ever sent that message. Same for superluminal warp drives and wormholes.
Now, most physicists don't take these situations to be something we'll ever observe in our world. But tachyons from Special Relativity or closed-timelike-curves in General Relativity are not contradictory: one can use the math to produce these situations.
Epistemic possibility is relevant here, since I'm relying on mathematical possibilities from physics. Given what we know, tachyon particles are not likely to occur if ever. Quantum Field Theory (QFT) predicts that a quantum field with imaginary mass never produces tachyon particles, but instead represents an unstable vacuum. Superluminal warp drives and wormholes are different. Nothing outright forbids such phenomena (yet), but they rely on negative mass-energy in astronomical quantities (which has never been observed).
Important to note here, is that even if someday we learn that tachyon particles and CTCs are forbidden by the laws of physics, their logical possibility is secured by the consistency of the math that allowed us to even hypothesize them.
So to answer your question: no, they are not logically impossible, since the math permits them in both SR and GR. We are getting a picture from QFT that at least tachyon particles will probably never be observed. Closed-timelike-curves are an open question: we'll need a theory of quantum gravity to get a definitive answer on that. Yet, even if that happens, it just means our universe won't have these phenomena (logically possible, but physically impossible).