Considering two random variables $X$ and $Y$.
I have seen stated in a number of places that $\frac{\overline{x}}{\overline{y}}$ is a consistent estimator for $E(\frac{x}{y})$ "under weak assumptions".
What assumptions are necessary?
Any proofs of the consistency would also be greatly welcomed.
It's not generally true; it's not clear that any set of assumptions that could reasonably called weak will do to make it true.
e.g. consider an extremely simple example -- iid $X$ and $Y$ taking the values $1$ and $2$ with equal probability.
Then as $n\to\infty$, $\bar{X}$ and $\bar{Y}$ converge to a constant, $1.5$ and the ratio will converge to $1$ via Slutsky.
However $E(\frac{X}{Y})$ is $\frac98$.
Note that for $Y$ a positive variate with nonzero variance, $E(\frac{1}{Y})>\frac{1}{E(Y)}$. Consequently, when $X$ has positive expectation, $X$ and $Y$ are independent and $Y$ is positive and has nonzero variance, $E(\frac{X}{Y})>\frac{E(X)}{E(Y)}$.
I don't think any set of conditions I'd reasonably call weak would exclude all of those cases (and that's just a subset of the cases where it's not going to work).