Let $f: A \times B \rightarrow C$ be continuous and closed under product of closed subsets of A and B, is $f$ closed? Assume product topology on $A \times B$. To make clear th title:
$f$ is a countinuous map such that if $R \subset A$ and $S \subset B$ are closed sets, then $f(R \times S) \subset C$ is a closed set of $C$. 
Question: if $T$ is any closed subset of $A \times B$, is $f(T) \subset C$ closed ?

Not in general.
In this context let $f:\mathbb R^2\rightarrow\mathbb R$ be prescribed by $\langle x,y\rangle\mapsto x$, let $\mathbb R$ be equipped with its usual topology, and $\mathbb R^2$ with the product topology.
Then $f$ is continuous and $f(R\times S)=R$ for $R,X\subseteq\mathbb R$, so the conditions mentioned in your question are satisfied.
The set $T:=\{\langle x,y\rangle\mid x>0\wedge y\geq\frac1{x}\}\subset\mathbb R^2$ is closed.
However, the set $f(T)=(0,\infty)\subset\mathbb R$ is not closed.
Proved is actually that projections are not necessarily closed maps.
