Actually, $\overline{f(Y)}\cap H$ and $\overline{f(Y\cap Z)}$ don't even have to be of the same dimension:

Let $X=Y=\mathbb A^2$ with coordinates $x,y$ and $f:X\to \mathbb P^2$ the morphism $(x,y)\mapsto [x:xy:1]$. Further let $x_0,x_1,x_2$ denote the homogenous coordinates on $\mathbb P^2$ and let $H=Z(x_0)$. Then $Z=f^{-1}H=Z(x)\subset Y=\mathbb A^2$.
Now, as you observe, $f(Y\cap Z)=f(Y)\cap H=\{[0:0:1]\}$, a single (closed!) point and hence the same holds for its closure: $\overline{f(Y\cap Z)}=\{[0:0:1]\}$. At the same time, $f$ is clearly dominant, i.e., $f(Y)$ is dense in $\mathbb P^2$ and hence $\overline{f(Y)}\cap H=H$.

The problem is that quasi-projective varieties are *missing* some pieces. In this example if you compactify $Y$ to start with, and resolve the indeterminacies of the morphism, then the image is the entire $\mathbb P^2$ and everything is dandy.
Of course, if you assumed that $Y$ was projective, then the statement would be trivially true, since in that case $f(Y)$ is closed.

To salvage the situation and get a condition for a quasi-projective variety to have this condition, you can do the following:

In addition to your setup, assume that:

$\bullet$ $f(Y)\subset \mathbb P^n$ is a quasi-projective variety. (I.e., it's open in its closure).

$\bullet$ $H\cap \overline{f(Y)}$ is irreducible (you can take this instead of assuming that $Y\cap Z$ is irreducible).

In this case it follows that $f(Y\cap Z)=f(Y)\cap H=f(Y)\cap \big(H\cap \overline{f(Y)}\big)$ is a non-empty open subset of the irreducible set $H\cap \overline{f(Y)}$ and hence it is dense in it.

The above example shows that the first condition ($f(Y)$ being quasi-projective) is necessary and Dustin's example shows that the second ($H\cap \overline{f(Y)}$ being irreducible) is. This seems to suggest that this statement is the best you can hope for. (Actually, instead of irreducibility in the second condition you could assume that $f(Y)$ intersects non-trivially all the components of $\overline{f(Y)}\cap H$, but that seems harder to check).