I came across the following conjecture (?) regarding (non-isolated) singularity in complex analysis: Suppose we have a function of a complex variable $f(z)$, which has poles at a set of points $\{z_n\},\ n=1,2,3, ...$ with a limit point $z_0 = \lim_{n\rightarrow \infty }z_n < \infty$. Then, $z_0$ must also be singular. My conjecture is that $z_0$ is a essential singularity.
One illustration of this is the function $f(z) = \frac 1 {e^{1/z}-1}$, which has simple poles at $z=\frac 1 {2\pi i n}$ ($n\in Z$) with residues $4\pi^2 n^2.$ This function has a essential singularity at $z=0$.
Do you think this is correct? If yes, is there any theorem regarding this?
No, it is not correct. The concept of essential singularity is defined only for isolated singularities (as are the concepts of removable singularity and of pole). So, if a singularity is not isolated, it makes no sense to ask which type of singularity it is: removable, pole or essential.
In particular, it is not true that $0$ is an essential singularity of the function from your example.