1. Forster just touches the Riemann-Hilbert problem and fiber bundles. Expansion on this can be interesting
I recommend the books of Bolibrukh. 

2. Applications of compact Riemann surfaces to solitons ("Explicit solutions" of the Koreweg-de-Fries equation etc. In a comprehensive course of algebraic curves and Riemann surfaces taught by Drinfeld, that I took in early 1980-s this was included as an example of application. (Exposition was based on Krichever's papers which were new at that time. Now you can find this in many books). 

3. Belyi theorem was used in this course as a HW exercise, but since then much interesting stuff was added to this.

4. Myself, I use holomorphic dynamics to "spice" my Riemann surface courses,  also Kleinian groups. Especially Sullivan's proofs of the finiteness
and non-wandering theorems. 

5. Theta-functions and the explicit solution of the inversion problem are not mentioned
in Forster, though this material is due to Riemann himself. This in turn has a lot of applications, in particular, to item 2 above. On this I recommend Mumford's 
classical Tata lectures on Theta.