You cannot (or at least, not without phenomenal effort, in the best of cases) prove a negative existence statement in science, i.e. find something that will tell you, with certainty, that some "whatever" does not exist. Not with tachyons, and not with Bigfoot, either.
The evidence that they don't exist is simply that in all attempts to find one, we never have. You may have heard that "absence of evidence is not evidence of absence" - but absence of evidence where we might expect to find evidence can indeed be evidence (not proof) of absence. Of course, we also don't know of all places where we might find evidence - so again, the strength of our "they don't exist" claim likewise has to be circumscribed there, too. Just like with Bigfoot - there's enough people in the woods that, since such are an environment where you'd expect to find them, the failure to observe bones likewise constitutes empirical evidence in the same way that it does not exist. But it is always - always possible whatever you're after exists in potentially any places you have not looked.
Theoretical arguments about causality are not disproofs - theory could be wrong. But the fact nobody has sent us a message from the future in any incident that passes scientific muster is, like above, also countermanding evidence. That said, there are ways tachyons could exist in which they would also be prevented from retrocausal action - one possibility is that absolute relativity is wrong instead, and there is a preferred rest frame (which might only be detectable using tachyons). Instead of considering them "disproofs", I'd actually consider them explanations for why that tachyons "don't exist" in the empirical sense mentioned above. There are many ways that our world could be put together that resolve these causality issues - but the easiest one is simply that the relevant retrocausal phenomena just don't exist to begin with.
