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JGWeissman comments on Causality does not imply correlation - Less Wrong
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This is the right idea. For small epsilon, B(t) should have a weak negative correlation with A(t - epsilon), a weak positive correlation with A(t + epsilon). and a strong positive correlation with the difference A(t + epsilon) - A(t - epsilon).
The function A causes the function B, but the value of A at time t does not cause the value of B at time t. Therefore the lack of correlation between A(t) and B(t) does not contradict causation implying correlation.
Only trivially. Since B = dA/dt, the correlation between B and dA/dt is perfect. Likewise for any other relationship B = F(A): B correlates perfectly with F(A). But you would only compare B and F(A) if you already had some reason to guess they were related, and having done so would observe they were the same and not trouble with correlations at all.
If you do not know that B = dA/dt and have no reason to guess this hypothesis, correlations will tell you nothing, especially if your time series data has too large a time step -- as positively recommended in the linked paper -- to see dA/dt at all.