As R W noted, the relative entropy depends only on the individual (marginal) distributions of $X$ and of $Y$; it does not depend on the joint distribution of $X$ and $Y$. So, the condition that $Y$ is a function of $X$ is quite irrelevant as far as the relative entropy is concerned, not matter what the function is.
Perhaps you mean the conditional (rather than relative) entropy
$$H(Y|X):=-E\ln p(Y|X)$$
of $Y$ given $X$, where $p$ is the conditional probability mass function (pmf) of a random variable $Y$ given $X$ (assuming that the conditional distribution of $Y$ given $X$ is discrete).
If so, then (as you guessed in the original version of your post) $H(Y|X)=0$ if $Y$ is a function of $X$ , because then the conditional pmf $p$ is supported on a singleton set, for each given value of $Y$.