What does lack of evidence of a causal relationship tell you?

Yes, see this for details on why this is so.

But in probability theory, absence of evidence is always evidence of absence. If E is a binary event and P(H|E) > P(H), "seeing E increases the probability of H"; then P(H|~E) < P(H), "failure to observe E decreases the probability of H". P(H) is a weighted mix of P(H|E) and P(H|~E), and necessarily lies between the two.

The causality looks like something. An obscure common cause is the most obvious (to me) source of the correlation if no one's put forth a plausible causal relationship yet. I'm not sure what you mean, though, by "causal relationship between X and Y"... do you mean specifically a relationship of the form "X -> ..Z.. -> Y" / "X <- ..Z.. <- Y", or do you mean "any causal structure connecting X and Y in any way"?

(Are you interested in some specific X,Y but phrasing it generally so we don't get distracted? I feel like seeing some examples of the failed tests run by statistically competent scholars would help me know what they haven't ruled out)

If there is no correlation between how many apples I drop from a height and how many hits the bottom, then the most likely hypothesis is that either there is some kind of barrier and the dropped apples I'm noticing come from an unrelated source, or that some process is reacting to me dropping X apples and dropping k-x apples.