consequences of rejected/accepted hypothesis 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?
 A: 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).
A: 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). 
A: 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: 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.
