From the document I found here (and many others which cover this exact same problem as well), we have the following where I added some edits to help follow along:
So I went ahead and tested the Dirichlet BCs case for different points $(x_0,y_0)$ for Green's Function, and I do in fact get an error at the point where $x=x_0$ and $y=y_0$. Basically: $$\lim_{x_0\rightarrow x,y_0\rightarrow y} G(x_0,y_0) = -\infty$$ This problem looks familiar from my classes in calculus having to do with natural logs and derivatives at 0, but I don't know if that's what I'm looking for. Any advice?