All known spacetime shortcuts, such as wormholes, Alcubierre drive and Krasnikov tubes, admit some sort of CTC metric variation. As far as I know, it's not really a theorem, just that it applies to all those cases.
But those CTC metrics do not necessarily mean much, because they do not specify whether it is possible to evolve the spacetime shortcut (the term spacetime shortcut isn't really standard by the way but Krasnikov has used it here) to the point where it forms a closed timelike curve.
There are three main arguments forbidding compactly generated time machines (that is, where the time machine is not the entire universe). The main one is energy conditions, which all spacetime shortcuts already violate, so it will not help here. The second is stability : CTCs that exist tend to collapse if matter tries to cross it. But this issue does not exist for all CTCs. And the big one is that, in semi-classical approximations, the closer to a CTC one gets, the higher the stress energy tensor of the vacuum gets, due to quantum effects, making it (possibly) impossible to form.
So, if any of these old in all realistic cases (there are theoretical scenarios that violate all of them), you might be able to build spacetime shortcuts without violating causality. Trying to build a time machine will just make the whole thing collapse. You may not even need to try to build a time machine : wormholes get time shifted just by being in different gravitational potentials, which might make then naturally get more and unstable as time goes on.
