I am told that material implication does not always "work." I'm not sure what the means exactly. Can readers here give me "real-life" examples of it not working, e.g. logical propositions $A$ and $B$ such that $A$ is false and $B$ is true and the implication $A\implies B$ is false.
Here is one absolutely standard line of thought (essentially due to Ramsey, some 90 years ago).
Jill is a little late home. Thoughts about various unlikely catastrophes pop unbidden into Jack's mind, such as $P$: Jill has had an accident. But Jack is good at keeping his worries in check. He in fact thinks it very probable that not-$P$.
Now, Jack knows that a disjunction is at least as likely to be true as its first disjunct (put it this way: there are at least as many ways the world might go which make $A$ or $B$ true as make $A$ true). So rational Jack realizes it is also very probable that not-$P$ or $Q$, for any second disjunct at all, including e.g. $Q$: Jill has been trampled by a herd of elephants. Since he knows that $(P \to Q)$ is by definition just not-$P$ or $Q$, Jack will also therefore give a very high degree of credence to the material conditional $(P \to Q)$ because of his very high confidence in the truth of not-$P$.
However, living as they do in a small English town, Jack will think it is very improbable indeed that, if the worst has happened and Jill has had an accident, then she has been tramped by a herd of elephants. In other words, Jack will give a very low degree of credence to the conditional if P then Q.
But how can this be, on the view $if$ = $\to$? If ordinary conditionals are no more and no less than unadorned material conditionals, rational Jack should give the same level of credence to if P then Q as to $(P \to Q)$. Since Jack's very different levels of credence are perfectly rational, 'if's aren't '$\to$'s.
I have noted the following fallacies that can spin from having $A \Rightarrow B$ available, but then jumping to conclusions. The following are all fallacies, means they are not tautological hence generally valid, they have all counter examples in propositional logic:
Affirming a Disjunct
$(p \vee q) \wedge p \Rightarrow \neg q$
Affirming the Consequent
$(p \Rightarrow q) \wedge q \Rightarrow p$
Commutation of Conditionals
$(p \Rightarrow q) \Rightarrow (q \Rightarrow p)$
Denying a Conjunct
$\neg (p \wedge q) \wedge \neg p \Rightarrow q$
Denying the Antecedent
$(p \Rightarrow q) \wedge \neg p \Rightarrow \neg q$
$(p \Rightarrow q) \Rightarrow (\neg p \Rightarrow \neg q)$