Does anyone know an explicit, exact description of the eigenforms of the Laplacian on a nonflat twotorus?

This is not so much an answer as a few remarks and a caution. If I understand your request correctly, I think that it is unlikely that you are going to find a truly explicit example. First, let me remark that, since we are considering a compact, oriented surface $T^2$, it would be enough to know all of the eigenfunctions of the Laplacian on $0$forms, since, by the Hodge decomposition theorem, for positive eigenvalues $\lambda$, any eigenfunction of the Hodge Laplacian on $2$forms with eigenvalue $\lambda$ would be of the form $\ast f$ where $f$ is a $\lambda$eigenfunction, and any eigenform of degree $1$ with eigenvalue $\lambda$ would be of the form $df_0 + \ast d f_1$ for some $f_0$ and $f_1$ that are $\lambda$eigenfunctions. Second, if we assume that the metric is given in the form $g = F(z) dz\circ d\bar z$ where $F$ is a positive, $L$periodic function on $\mathbb{C}$, then this metric will have nonconstant Gaussian curvature if and only if $F$ is not constant. In any case, a $\lambda$eigenfunction will be an $L$periodic function $f$ on $\mathbb{C}$ that satisfies $$ 4f_{z\bar z} + \lambda F f = 0. $$ So you are asking for a method of explicitly describing all of the $L$periodic solutions of this equation. I am not aware of any positive, nonconstant $F$ for which this is known. If you go down a dimension and ask for the list of positive, $\pi$periodic functions $F$ on the real line for which all of the $\pi$periodic solutions of $$ f''(x) + \lambda F(x) f(x) = 0 $$ are explicitly known for each $\lambda>0$, I think you will find that this list is very short. It makes me suspect that, in the $2$dimensional case, the list is not longer. 

