Lower bound of real valued bounded function
You have to specify a loss -- say, $\ell_1$ for simplicity, so the risk of a hypothesis $h$ is $E|h(X)-Y|$. Then at the very least, by reduction to the VC case, to achieve accuracy $\epsilon$ you will need at least 
$$\Omega(d_F(\epsilon)/\epsilon^2)
\qquad (*)$$
examples, where $d_F(\epsilon)$ is the fat-shattering dimension of the class $F$ at scale $\epsilon$. Just take an $\epsilon$-shattered set of size $d_F(\epsilon)$ and put uniform point masses on the $x$'s, while setting the $y$'s to be $\pm\epsilon$ with appropriate probabilities (just as in the binary case -- which is carefully worked out in these notes:
https://www.cs.bgu.ac.il/~asml162/wiki.files/agnostic-pac-lb.pdf
).

I don't know if this is the tightest bound possible; the best upper bound I know, given here, 
https://www.sciencedirect.com/science/article/pii/S0022000097915579
behaves as
$$ O(\Omega(d_F(\epsilon/5)\log(1/\epsilon)/\epsilon^3.$$

Update.
The lower bound $(*)$ is for excess risk -- i.e., in the agnostic PAC setting. The realizable PAC lower bound can be straightforwardly be adapted to give a lower bound of $\Omega(d_F(\epsilon)/\epsilon)$ for the absolute risk.