ag.algebraic geometry - is there a push-forward of closed subschemes?

It sounds as though what you want is the closure of the image of $F$ under $f$. (That is, the minimal closed subscheme of $Y$ factoring $f$.)

If $X =$ Spec $A$, and $Y =$ Spec $B$, and $F =$ Spec $A/I$, and $f$ corresponds to the ring map $f':Bto A$, then we can consider the preimage $J$ of $I$ under $i'$. Consider the set of primes in $B$ containing $J$. Of course, any prime in the image of $F$ under $f$ must contain $J$, since its preimage under the ring map has to contain $I$. Thus, Spec $B/J$ contains the preimage. You can check that it's the biggest such ideal, noting that in order for us to have a map $B/J to A/I$, $J$ should be contained in the preimage of $I$.

Being the closure of the image of $F$ under $f$ is a universal property of sorts (in particular, it's unique), which more or less allows us to argue that this construction generalizes to non-affine schemes. (Just apply this local construction, and uniqueness tells us that the local constructions glue together.)