To determine if the relation $R$ is symmetric, we need to check if for every $(x, y)$ in $R$, the pair $(y, x)$ is also in $R$.
$R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$.
We have $(1, 3)$ in $R$, and $(3, 1)$ is also in $R$.
We have $(4, 2)$ in $R$, but $(2, 4)$ is also in $R$.
However, we have $(2, 3)$ in $R$, but $(3, 2)$ is not in $R$.
Since there exists a pair $(2, 3)$ in $R$ such that $(3, 2)$ is not in $R$, the relation $R$ is not symmetric.

Therefore, the correct answer is Option C: $R$ is not symmetric.