Where do theta functions and canonical Green functions appear in physics

In the beginning of Section 5 in his article, Wentworth mentions a result of Bost and proves it using the spin-1 bosonization formula. This result provides a link between theta functions, canonical Green functions, and Faltings' delta invariant on a compact Riemann surface of positive genus.

Are there any other situations in physics which involve canonical Green functions on compact Riemann surfaces?

Theta functions appear when determining various partition functions, e.g. the translational PF of an ideal gas

A rich source of applications can be found in string theory. Most straightforwardly, the partition function, or vacuum amplitude, of a conformal field theory on a Riemann surface are essentially given by theta functions (and their generalizations, e.g. Siegel modular forms). Various theta function identites have important physical consequences.

The simplest case to look at would be the torus, discussed at length for example in Polchinski's textbooks - the four spin structures on the torus lead to the four Jacobi theta functions, and the Green's function can also be expressed in terms of theta functions as a consequence. (By the way, bosonization also appears in this context as the worldsheet of a string is 1+1 dimensional (see e.g. Green-Schwarz-Witten's books)).

On the other hand, a very active recent topic would be the occurrence of mock modular forms and the "wall-crossing" phenomenon.

The Greens function for a charged particle moving on a circle which encloses a nonzero magnetic flux is given by the Jacobi theta function. The relation between the expression one obtains using canonical or path integral methods is given by the $\tau \rightarrow -1/\tau$ modular transformation of the theta function.

Theta functions also appear in the treatment of the quantum Hall effect on the torus, see F.D.M Haldane and E.H.Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect, Physical Review B, Vol 31, Number 4, 1985 .

I think an example of the kind of result you're looking for might be found in the paper On the Thomae Formula for ZN Curves. Although it proves a result purely about theta functions the proof is derived from work investigating Green's functions arising from conformal field theories on Riemann surfaces with $Z_N$ symmetry.