Here's a simple example. Let $X=\mathbb{N}^2\cup\{\infty\}$ topologized such every subset of $\mathbb{N}^2$ is open and the neighborhood filter at $\infty$ is generated by the sets $\mathbb{N}\times[n,\infty)\cup\{\infty\}$ for $n\in\mathbb{N}$. Let $Y=\{0,1\}$ topologized such that $\{1\}$ is open but $\{0\}$ is not. We identify $Z=Cont(X,Y)$ with the set of open subsets of $X$. For each $x\in X$, let $D_x=\{U\in Z:x\in U\}$, and for each $n\in\mathbb{N}$ let $E_n=\{U\in Z:\mathbb{N}\times[n,\infty)\cup\{\infty\}\subseteq U\}$. For each $N\in\mathbb{N}$, let $\tau_N$ be the topology generated by the $D_x$ for all $x\in X$ and the $E_n$ for all $n\geq N$.

I claim that each $\tau_N$ is admissible. Indeed, the fact that $D_{(m,n)}\in\tau_A$ implies that evaluation is continuous at any point of the form $(U,(m,n))\in Z\times X$, and the fact that $E_n\in\tau_A$ implies that evaluation is continuous at any point of the form $(U,\infty)\in Z\times X$ such that $\mathbb{N}\times[n,\infty)\cup\{\infty\}\subseteq U$. Since every neighborhood of $\infty$ contains $\mathbb{N}\times[n,\infty)$ for some $n\geq N$ and evaluation is automatically continuous at $(U,\infty)$ if $\infty\not\in U$, we conclude that evaluation is continuous everywhere.

Now let $\tau$ be the intersection of all the $\tau_N$. I claim that $\tau$ is not admissible. Indeed, suppose evaluation was continous with respect to $\tau$ at the point $(X,\infty)\in Z\times X$. This means that there is a neighborhood $V\subseteq X$ of $\infty$ and a $\tau$-neighborhood $F$ of $X$ such that for all $W\in F$, $V\subseteq W$. That is, there is an $n\in\mathbb{N}$ such that $F\subseteq E_n$. But for any $N>n$, it is easy to see that there is no nonempty $F\in\tau_N$ such that $F\subseteq E_n$.

With some work, this argument can be generalized to apply to any $X$ which is Hausdorff and not locally compact. In full generality, it can be shown that there is a smallest admissible topology on $Cont(X,Y)$ (for $Y$ the Sierpinski space, or equivalently for all $Y$) iff $X$ is core-compact. There is a nice brief overview of this (including the definition of core-compact) on nLab; full details can be found in this nice little paper.