Take, for example, $\Lambda=\{(n,m):n,m \in \mathbb{Z}\}$ and look at the sum for $z=0$ -- ignore $\omega=(0,0)$ for the sake of this discussion. Then you can rearrange terms so that you sum over the terms which are from the squares centered around $0$. The inner square has $8$ points contributing to the sum the next one already $16$, in general the $n$- th will contribute $2*(2n+1)+ 2*(2n-1)$ (draw a picture). So each of these squares contributes $Cn$ points, but the order of the terms is roughly $1/n^2$. In other words, the sum behaves like a harmonic series. This is basically because the number of terms of a certain magnitude is increasing, due to 2 dimensions, linearly. In one dimension the number of terms of a certain magnitude is constant.
This behaviour is similar to the behaviour you observe if you try to integrate $\int_{|x|>1}\left(\frac{1}{|x|}\right)^k$ in spaces of increasing dimensions. The bound for $k$, for which this converges, depends in a similar manner (for the same reason) on the dimension $n$. Technically this corresponds to a factor of $r^{n-1}$ in the volume element if written down in polar coordinates, geometrically this of course just the way the surface of the $(n-1)$-dimensional sphere scales.
In other words, this is not a question about complex analytic function. It becomes a question about these if you add terms forcing normal convergence and arrive at the Weierstrass function. But then other effects become predominant (topological ones, elliptic functions are functions on a torus).