Let's say I'm running a linear regression that has the form $y = \beta_0 + \beta_1A+\beta_2B+\beta_3AB +\epsilon$.
If $\beta_3$ is positive, does this imply a positive correlation between $A$ and $B$? (Conversely, a negative correlation if $\beta_3$ is negative?)
No, a non-zero $\beta_3$ does not imply $A$ and $B$ are correlated. It implies $y$ is correlated with $AB$.
Imagine we have data on visits by people to a gas station.
$A \cdot B$ is how much it would cost to fill the person's gas tank. $AB$ is almost certainly correlated with $y$, the spending on gas this visit.
A positive $\beta_3$ in this trivial example does not imply that the size of someone's gas tank is correlated with the price of gas. A positive $\beta_3$ would mean that spending $y$ is positive related to the carrying capacity of someone's gas tank measured in dollars (i.e. $AB$).
Here is a potential applied counterexample: suppose $A$ is gender, $B$ are years of schooling and $y$ are labor-market earnings. So, after, say, 12 years of primary and secondary school and a three-year Bachelor degree, you would have completed 15 years of schooling.
Then, it is not totally off to assume that $A$ and $B$ are uncorrelated - in the past, men used to have higher degrees, nowadays, if anything, women. So there probably was a moment in the (not so distant) past when gender and years of schooling were uncorrelated, and the correlation certainly is not strong today.
And yet, it is not difficult to make a case that $\beta_3\neq0$, as an additional year of schooling may have a differential effect on earnings for men than for women.
This would, for example, be the case when there is wage "discrimination" (in quotation marks as it is a hotly debated issue) mostly in jobs for more highly educated employees. Anecdotical evidence suggests that this may be the case, as male executives tend to be better paid than female ones. On the other hand, salaries in jobs that require less education may be more frequently determined by broad agreements between unions and employers' associations (at least in, for example, continental Europe), leaving less room for wage discrimination.
(The quotation marks could for example be justified by the fact that this simple story does not account for sectors, experience, etc.)
