Here's an idea. Not a particularly good one, but it's too long for a comment.
Using summation formuli you can rewrite the term in your sum as a polynomial in $\cos(\theta_i)$ and $\sin(\theta_i)$ for $i=1,\ldots,n$. Then you can rationally parametrize your circles by $\cos(\theta_i) = \frac{1-t_i^2}{1+t_i^2}$, $\sin(\theta_i) =\frac{2t_i}{1+t_i^2}$ and obtain a sum of rational functions thus reducing the problem to a system of polynomial equations. (The last condition $\sum_i \theta_i = 0$ just reduces the number of your variables but you have to pay the price of much uglier expressions coming from $\sin$ containing say $\theta_n = -\sum_{i\neq n} \theta_i$.)
For fixed $n$ you may (hopefully) deduce finiteness of the solution set via Groebner basis.
Or maybe there's even better coordinate system for the cut of your torus (which is, if I am not mistaken, topologically just the torus of dimension one less $\simeq \mathbb{T}^{n-1}$) and the functions in your equations have some "geometrical meaning".