Here’s a small sample of examples.
The $G$ graph of $y=1/x$ is closed in $\Bbb R^2$, and the map $p:\Bbb R^2\to\Bbb R:\langle x,y\rangle\mapsto x$ is continuous, but $p[G]=(\leftarrow,0)\cup(0,\to)$, which is not closed in $\Bbb R$.
The map $\Bbb R\to\Bbb R:x\mapsto e^{-x}$ sends the closed subset $[0,\to)$ of $\Bbb R$ to the non-closed subset $(0,1]$. Other functions with horizontal asymptotes provide similar examples.
If $X$ is any non-closed subset of a space $Y$, the inclusion map $i:X\to Y:x\mapsto x$ gives a trivial example, since $X$ is a closed subset of itself.
Another trivial example is obtained by taking any infinite set $X$, letting $\tau_d$ be the discrete topology on $X$, and letting $\tau$ be any other topology on $X$. The identity map from $\langle X,\tau_d\rangle$ to $\langle X,\tau\rangle$ is automatically continuous. However, there is at least one $x_0\in X$ such that $\{x_0\}\notin\tau$ (i.e., $x_0$ isn’t an isolated point of $\langle X, \tau \rangle$); if $A=X\setminus\{x_0\}$, then $A$ is closed in $\langle X,\tau_d\rangle$ (as is every subset of $X$), but $A$ is not closed in $\langle X,\tau\rangle$.