A and B are some statements such that A implies B. I test the null hypothesis that A is true. If my test fails to reject A, does that result say anything about B? Analogously, if instead I test the null hypothesis that B is true, and my test rejects B, can I conclude that A is rejected as well?
I refer to my answer to What follows if we fail to reject the null hypothesis?, because it is also a matter of power of the test.
In logic, if $A \implies B$, then the being true of B does not lead to anly conclusion on A, however, if B is false, then A can not be true (because if A would be true, then, by modus ponens, B would be true but as B is false this can not be). So $(A \implies B) \iff (not(B) \implies not(A))$.
As said these are rules in logic, where A is either true or false, in statistical hypothesis testing we have no certainty (else we do not do statistics) so it is different (except in the unrealistic case where tests have a power of 100pct).
Applying straightforward logic, A => B can be translated to ¬A or B, which in the end means that if B is rejected, so is A.
As f coppens's answer says, we can never be 100% sure because in statistics we are always working with probabilities. I would say these logic rules follow, but they are modified by the degree of certainty you are working with. If B is rejected, then there is a similar level of certainty about rejecting A. Also, failing to reject A does not enable you to accept it, nor to say anything about B.
Nonetheless, I would point out this: when A and B are real statements about your field of study, you should be very careful about how sure you are that A => B. This is a very strong assumption that could lead you to wrong conclusions if this assessment is not entirely true.
A implies B, understood as logical implication, means that if A is true, then B is true. However if A is false, this says nothing about B, and if B is true, this says nothing about A.
According to that definition, concluding that B is true will shed no light over A. Also, concluding that A is false will give you no information about B.
Finally, if you conclude that A is true, then you are safe say that B is true, though this conclusion should not be reached from failing to reject the null, since this doesn't mean that the null is true (see this for more information).
At least in the frequentist school of statistical inference, failing to reject a hypothesis is not sufficient to prove it -- that would typically require rejecting all other plausible hypotheses.
In your counterexample, if you are certain that A implies B, then a rejection of B could be interpreted as a rejection of A. However, that would depend entirely on the strength of that relationship.